Is there such a thing as a matched-pairs t-test between two lists of proportions? I am aware that there is a matched-pairs t-test between two lists of counts. However, I cannot find any matched-pairs t-test of proportions. For example, list a may be [1/2, 24/30, 5/8] and list b may be [51/100, 4/9, 4/8].
If not, how can I go about comparing two lists like this?
 A: I'd go with a Wilcoxon signed ranks test. The difference between paired proportions is arguably not normally distributed (I'm not sure about that), but that doesn't matter if the test is non-parametric. Specifically, the distribution of the differences should be symmetrical around their median. 
As long as the pairs are independent from each other, the assumptions of the test are probably fulfilled in your case. 
Very likely, t-test would give a similar answer because it is generally thought to be robust (but see here).
Alternatively, you can always use a randomization procedure or a permutation test. Crunching replicates with a computer can often substitute having an exact solution. In your case, I'd turn to the function permtest from the R package BHH2.
A: To expand on my comment:

I am aware that there is a matched-pairs t-test between two lists of counts. 

Matched pairs of counts might be tested in other ways. Once issue with using a paired t-test on counts is that the assumption of equal variance for observations (and even of differences) will almost certainly be violated.

However, I cannot find any matched-pairs t-test of proportions. For example, list a may be [1/2, 24/30, 5/8] and list b may be [51/100, 4/9, 4/8].

Well, those are also counts, but the usual assumption would be that they're binomial.
If the null is that within the pairs the proportions are equal (but not necessarily across pairs), then one approach would be to do a chi-square test. Each matched pair gives you a 2x2 table:
         Table 1:
        Gp1  Gp2    Total
   S     1   51       52
   F     1   49       50 

Total    2  100      102

A: I know this is an old question, but recently I had to run a similar analysis, and it turns out that the right approach is to use Cochran–Mantel–Haenszel test, see e.g. article in wikipedia.
