Program equivalence wherein the programs are known to always halt
As a counter-example to this, consider the Context-Free Equivalence problem: it's undecidable to determine, given two context free languages, whether they accept the same set.

If your problem were decidable, we could use it to determine CFL equivalence, since it's always possible to turn a CFL into an always-halting Turing machine.

So even for countably infinite inputs, the problem is undecidable.

It's also worth mentioning that, for standard Turing Machines, all input domains are countably infinite, since they're sets of finite strings.