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As Amit points out in his comment, if the Continuum Hypothesis holds in the ground model $V$, then the collection of ground model reals becomes countable in the extension $V[G]$ in which $\omega_1$ is collapsed, and therefore there must be a real in $V[G]$ dominating every real in $V$. More generally, consider the dominating number, which is the size of the smallest family of functions such that every function is dominated by a function in the family.

Theorem. The following are equivalent:

1. There is a forcing extension $V[G]$ collapsing $\omega_1$, but not adding a real dominating every ground model real.
2. The dominating number of $V$ is at least $\omega_2$.

Proof. Amit's CH observation generalizes to show 2 implies (the contrapositive of) 1 implies 2, because if $V$ has a dominating family of size $\omega_1$, then this family becomes countable in any $V[G]$ collapsing $\omega_1$, in which case there is a function in $V[G]$ dominating it, and hence dominating any ground model function.

Conversely, suppose that the dominating number is at least $\omega_2$. Now, consider the forcing to collapse $\omega_1$ to $\omega$ by finite conditions. This forcing has size $\omega_1$. Suppose that the forcing adds a function $f={\dot f}_G:\omega\to\omega$ dominating every function in $V$. For any ground model function $h:\omega\to\omega$, there is a condition $p_h$ and a number $n_h$ such that $p_h$ forces that $\dot f$ dominates $h$ beyond $n_h$. Since there are only $\omega_1$ many conditions in the forcing, it must be that some condition $p$ and natural number $n$ works for an unbounded family of functions $h$, since if each such family was bounded in $V$, then we would be able to form a dominating family in $V$ of size $\omega_1$, contrary to hypothesis. So fix $p$ and $n$ such that there is an unbounded family $F$ of functions $h$ such that $p$ forces $\dot f$ dominates $h$ beyond $n$. Now define a function $f^*(m)$ by choosing any condition (the first with respect to some well-ordering) $p_m$ extending $p$, such that $p_m$ decides the value of $\dot f(\check m)$. Since this must be larger than $h(m)$ for any $h\in F$, when $m\gt n$, it follows that $F$ was not unbounded after all, a contradiction. QED

As Todd mentions below, the idea of the proof easily generalizes to larger cardinals. Specifically, no forcing notion of size less than $\frak{d}$ can add a dominating real. So when $\frak{d}$ is very large, you can also collapse $\omega_2$ and larger cardinals without adding a dominating real.

In particular, since it is known to be consistent that the dominating number can be large, the answer to the title question is yes, it is consistent that this can happen.

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As Amit points out in his comment, if the Continuum Hypothesis holds in the ground model $V$, then the collection of ground model reals becomes countable in the extension $V[G]$ in which $\omega_1$ is collapsed, and therefore there must be a real in $V[G]$ dominating every real in $V$. More generally, consider the dominating number, which is the size of the smallest family of functions such that every function is dominated by a function in the family.

Theorem. The following are equivalent:

1. There is a forcing extension $V[G]$ collapsing $\omega_1$, but not adding a real dominating every ground model real.
2. The dominating number of $V$ is at least $\omega_2$.

Proof. Amit's CH observation generalizes to show 2 implies 1, because if $V$ has a dominating family of size $\omega_1$, then this family becomes countable in any $V[G]$ collapsing $\omega_1$, in which case there is a function in $V[G]$ dominating it, and hence dominating any ground model function.

Conversely, suppose that the dominating number is at least $\omega_2$. Now, consider the forcing to collapse $\omega_1$ to $\omega$ by finite conditions. This forcing has size $\omega_1$. Suppose that the forcing adds a function $f={\dot f}_G:\omega\to\omega$ dominating every function in $V$. For any ground model function $h:\omega\to\omega$, there is a condition $p_h$ and a number $n_h$ such that $p_h$ forces that $\dot f$ dominates $h$ beyond $n_h$. Since there are only $\omega_1$ many conditions in the forcing, it must be that some condition $p$ and natural number $n$ works for an unbounded family of functions $h$, since if each such family was bounded in $V$, then we would be able to form a dominating family in $V$ of size $\omega_1$, contrary to hypothesis. So fix $p$ and $n$ such that there is an unbounded family $F$ of functions $h$ such that $p$ forces $\dot f$ dominates $h$ beyond $n$. Now define a function $f^*(m)$ by choosing any condition (the first with respect to some well-ordering) $p_m$ extending $p$, such that $p_m$ decides the value of $\dot f(\check m)$. Since this must be larger than $h(m)$ for any $h\in F$, when $m\gt n$, it follows that $F$ was not unbounded after all, a contradiction. QED

Since

As Todd mentions below, the idea of the proof easily generalizes to larger cardinals. Specifically, no forcing notion of size less than $\frak{d}$ can add a dominating real. So when $\frak{d}$ is very large, you can also collapse $\omega_2$ and larger cardinals without adding a dominating real.

In particular, since it is known to be consistent that the dominating number can be large, the answer to the title question is yes, it is consistent that this can happen.

1

As Amit points out in his comment, if the Continuum Hypothesis holds in the ground model $V$, then the collection of ground model reals becomes countable in the extension $V[G]$ in which $\omega_1$ is collapsed, and therefore there must be a real in $V[G]$ dominating every real in $V$. More generally, consider the dominating number, which is the size of the smallest family of functions such that every function is dominated by a function in the family.

Theorem. The following are equivalent:

1. There is a forcing extension $V[G]$ collapsing $\omega_1$, but not adding a real dominating every ground model real.
2. The dominating number of $V$ is at least $\omega_2$.

Proof. Amit's CH observation generalizes to show 2 implies 1, because if $V$ has a dominating family of size $\omega_1$, then this family becomes countable in any $V[G]$ collapsing $\omega_1$, in which case there is a function in $V[G]$ dominating it, and hence dominating any ground model function.

Conversely, suppose that the dominating number is at least $\omega_2$. Now, consider the forcing to collapse $\omega_1$ to $\omega$ by finite conditions. This forcing has size $\omega_1$. Suppose that the forcing adds a function $f={\dot f}_G:\omega\to\omega$ dominating every function in $V$. For any ground model function $h:\omega\to\omega$, there is a condition $p_h$ and a number $n_h$ such that $p_h$ forces that $\dot f$ dominates $h$ beyond $n_h$. Since there are only $\omega_1$ many conditions in the forcing, it must be that some condition $p$ and natural number $n$ works for an unbounded family of functions $h$, since if each such family was bounded in $V$, then we would be able to form a dominating family in $V$ of size $\omega_1$, contrary to hypothesis. So fix $p$ and $n$ such that there is an unbounded family $F$ of functions $h$ such that $p$ forces $\dot f$ dominates $h$ beyond $n$. Now define a function $f^*(m)$ by choosing any condition (the first with respect to some well-ordering) $p_m$ extending $p$, such that $p_m$ decides the value of $\dot f(\check m)$. Since this must be larger than $h(m)$ for any $h\in F$, it follows that $F$ was not unbounded after all, a contradiction. QED

Since it is known to be consistent that the dominating number can be large, the answer to the title question is yes, it is consistent that this can happen.