Yes, there are several. Here’s a few which I personally care about (described in varying amounts of precision). This is not meant to be an exhaustive list, and reflects my own biases and interests.
I am focusing here on questions which have been open for a long amount of time, rather than questions which have only recently been raised, in the hopes that these are more easily understood.
MODEL THEORY
The compactness and Löwenheim–Skolem theorems let us completely classify those sets of cardinalities of models of a first-order theory; that is, sets of the form $$\{\kappa: \exists \mathcal{M}(\vert\mathcal{M}\vert=\kappa, \mathcal{M}\models T)\}.$$ A natural next question is to count the number of models of a theory of a given cardinality. For instance, Morley’s Theorem shows that if $T$ is a countable first-order theory which has a unique model in some uncountable cardinality, then $T$ has a unique model of every uncountable cardinality (this is all up to isomorphism, of course).
Surprisingly, the countable models are much harder to count! Vaught showed that if $T$ is a (countable complete) first-order theory, then — up to isomorphism — $T$ has either $\aleph_0$, $\aleph_1$, or $2^{\aleph_0}$-many countable models. Vaught’s Conjecture states that we can get rid of the weird middle case: it’s either $\aleph_0$ or $2^{\aleph_0}$. In case the continuum hypothesis holds, VC is vacuously true; but in the absence of CH, very little is known. VC is known for certain special kinds of theories (see e.g. Vaught's conjecture for partial orders and http://link.springer.com/article/10.1007%2FBF02760651) and a counterexample to VC is known to have some odd properties, including odd computability-theoretic properties (https://math.berkeley.edu/~antonio/papers/VaughtEquiv.pdf), but the conjecture is wide open.
NOTE: VC can be rephrased as a “countable/perfect” dichotomy, in which case it is not trivially true if CH holds and is in fact forcing invariant; see e.g. How do we know if Vaught's Conjecture is Absolute?
PROOF THEORY
If $T$ is a strong enough reasonable theory, we can define the proof-theoretic ordinal of $T$; roughly, how much induction is necessary to prove that $T$ is consistent. For instance, the proof-theoretic ordinal of $\mathrm{PA}$ is $$\epsilon_0=\omega^{\omega^{\scriptstyle\omega^{\scriptstyle\cdot^{\scriptstyle.{\cdot}}}}}.$$ Proof-theoretic ordinals have been calculated for a variety of systems reaching up to (something around) $\Pi^1_2$-$\mathrm{CA}_0$, a reasonably strong fragment of second-order arithmetic which is in turn a very very small part of ZFC. It seems unfair, based on this, to list “finding the proof-theoretic ordinal of ZFC” as one of these problems, based on how far away it is; but “find ordinals for stronger theories” is an important program.
See e.g. Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?
COMPUTABILITY THEORY
I believe the oldest open problem in computability theory is the automorphism problem. In Turing’s 1936 paper, he introduced — in addition to the usual Turing machine — the oracle Turing machine (or o-machine). This is a Turing machine which is equipped with “extra information” in the form of a (fixed arbitrary) infinite binary string. Oracle machines allow us to compare the non-computability of sets of natural numbers: we write $A\le_T B$ if an oracle machine equipped with $B$ can compute $A$. This yields a partial ordering $\mathcal{D}$, the Turing degrees. Initially the Turing degrees were thought to be structurally simple; for instance, it was conjectured (I believe by Shoenfield) that the partial order is “very homogeneous” (there were many different conjectures).
As it turned out, however, the exact opposite happens: the Turing degrees have surprisingly rich structure. See e.g. http://www.jstor.org/stable/2270693?seq=1#page_scan_tab_contents for an early example of this by Feiner, and http://www.pnas.org/content/76/9/4218.full.pdf for a later one by Shore. Indeed, currently the general belief is that $\mathcal{D}$ is rigid, and it has been shown (see e.g. https://math.berkeley.edu/~slaman/papers/IMS_slaman.pdf, Theorem 4.30) that $\operatorname{Aut}(\mathcal{D})$ is at most countable. The automorphism problem is exactly the question of determining $\operatorname{Aut}(\mathcal{D})$; I don’t have a reference as to when it was first stated, but I vaguely recall the date 1955.
We can also ask about “local” degree structures — e.g., the partial order of the c.e. degrees, or the degrees below $0'$ — and there are interesting connections between the local and global pictures.
Another structural question about the Turing degrees is what sort of natural operations on Turing degrees exist. For instance, there is the Turing jump, and its iterates; but these seem to be the only natural ones. Martin’s conjecture states that indeed, every “reasonable” increasing function on the Turing degrees is “basically” an iterate of the Turing jump; MC has a few different forms, for instance “all Borel functions …” or “In $L(\mathbb{R})$ …”. See e.g. https://math.berkeley.edu/~slaman/talks/vegas.pdf.
SET THEORY
An important theme in set theory is the development of canonical models for extensions of ZFC. The first example is Gödel’s $L$, which has a number of nice properties: a well-understood structure, a “minimality” property, and a canonical (in particular, foring-invariant) definition. We can ask whether similar models exist for ZFC + large cardinals: e.g. is there a “core” model for ZFC + “There is a measurable cardinal”? This is the inner model program, and has been developed extensively. Surprisingly, there is an end in sight: in an appropriate sense, if a canonical inner model for ZFC + “There is a supercompact cardinal” can be constructed, then this inner model will in fact capture all the large cardinal properties of the universe.
I am breezing past a truly gargantuan amount of detail here, but the picture is roughly accurate. See e.g. http://www.math.uni-bonn.de/ag/logik/events/young-set-theory-2011/Slides/Grigor_Sargsyan_slides.pdf for more details, as well as the recent presentation https://www.youtube.com/watch?v=MFDVN7UEUSg&list=PLTn74Qx5mPsQlRpBE5OnxMdN3R1d3DLUO&index=4 by Woodin.
FINITE MODEL THEORY
For a first-order sentence $\varphi$, let the spectrum of $\varphi$ be the set of sizes of finite models of $\varphi$: $$\operatorname{Spec}(\varphi)=\{n: \exists\mathcal{M}(\vert\mathcal{M}\vert=n, \mathcal{M}\models\varphi)\}.$$ We can ask what sets of natural numbers are spectra of sentences; in particular, the finite spectrum problem (see the really lovely paper Durand/Jones/Makowsky/More, Fifty years of the spectrum problem) asks whether the complement of a spectrum is also a spectrum. It is known, for example, that the complement of the spectrum of a sentence not using “$=$” is a spectrum (Ecsedi-Tóth, A partial solution of the finite spectrum problem).
There is a complexity theory connection here: a set is a spectrum iff it is in NE. So the finite spectrum problem asks, “Does $\mathrm{NE}=\mathrm{coNE}$?”
We can also ask about spectra for non-first-order sentences.
ABSTRACT MODEL THEORY
Abstract model theory is the study of logics other than first-order. The classic text is “Model-theoretic logics” edited by Barwise and Feferman; see (freely available!) https://projecteuclid.org/euclid.pl/1235417263. The field began (arguably) with Lindström’s Theorem, which showed that there is no “reasonable” logic stronger than first-order logic which satisfies both the Compactness and Löwenheim–Skolem properties.
Shortly after Lindström’s result, attention turned towards Craig’s interpolation theorem, a powerful result in proof theory (see https://math.stanford.edu/~feferman/papers/Harmonious%20Logic.pdf). Feferman, following Lindström, asked whether there is a reasonable logic stonger than first-order which satisfies compactness and the interpolation property. As far as I know, this question — and many weaker versions! — are still completely open.
I believe this is by far the youngest question in this answer.
