Assume you are given an explicit presentation of the $E_2$-terms of the Adams spectral sequence. Are the differentials on $E_2$ and further algorithmically computable? I do not care how practical it is, just if it can be done.
I have heard that the homotopy groups of finite CW complexes are algorithmically computable.
In principle, everything is algorithmically computable, but the proof does not lead to practical algorithms. In practice, you can find some differentials by ad hoc means and then deduce many more differentials. You can get some of these by just using the ring structure and the Leibniz property, but then you can get much further by using algebraic Steenrod operations. Bob Bruner has lots of relevant code at http://www.rrb.wayne.edu/papers/#code.
The work of Baues and collaborators on the secondary Steenrod algebra gives a category in which Ext is the $E_3$-term of the Adams spectral sequence. This must in some sense decide the $d_2$ differential, especially if there is a clear relation between Baues' E_3 = secondary-Ext and the usual $E_2$-term. I am sorry that I can't be more explicit off the top of my head.
As for Steenrod operation differentials, they only apply in $Ext(H^*R,F_p) \Longrightarrow \pi_*R$ for an $H_\infty$-ring spectrum $R$. Fortunately, the sphere spectrum is $H_\infty$ and perhaps this is what you care about. For such spectra, at $p=2$, the 'generic' differential is $d_2(\cup_i(x)) = h_0\cup_{i-1}(x)$ if $i \equiv n \pmod{2}$, and similarly at odd $p$. Unfortunately, 'most' elements are not in the image of a $\cup_i$ operation.
The computability result you have heard about is probably that of Ed Brown:
MR0083733
Brown, Edgar H., Jr.
Finite computability of Postnikov complexes.
Ann. of Math. (2) 65 (1957), 1--20.
As Neil says, it shows that the homotopy groups of a finite simplicial complex are effectively computable, but the result is not a practical algorithm. It also does not directly say anything about the Adams spectral sequence, which is a rather different approach to those homotopy groups.
