Did the big bang create an infinite number of photons?
As Ben Crowell already answered, the number of photons could be finite or infinite depending on whether the universe is finite. But I want to comment on the underlying assumption:
We will always be able to see the Cosmic Microwave Background at about [age of universe] light years away. Always.
It is possible that the expansion of the universe eventually stops then reverses and the unverse then recollapses, leading to a Big Crunch. This could happen, for example, if the universe is spatially closed, but there are other possibilities which result in Big Crunch. If that happens, the average density of matter would be increasing then and all the photons of the Cosmic Microwave Background would be eventually absorbed by heated (and opaque) matter. Of course, many new photons would be created at the same time, but those would not be the relics of the Big Bang.
If instead the expansion would continue indefinitely, then not only the number of CMB photons in a given constant volume would decrease in time, their wavelengths would also stretch, and this would mean that eventually it would not be possible to detect them. Assuming that the expansion would continue with approximately the same Hubble parameter, about every $10^{10}$ years the wavelength of CMB photon would grow by a factor of $e$. This means, that in $10^{30}$ years the wavelength of a typical CMB photon would be in excess of 10 light-years and in $10^{40}$ years wavelenth of CMB photons would exceed the size of de Sitter horizon, which also means that the CMB radiation by that time would be drowned in Gibbons–Hawking radiation coming from de Sitter horizon, and thus by then it would be impossible to detect CMB even in principle.
The cosmological evidence is currently consistent with either a closed or an open universe. A closed universe is spatially finite, has always been spatially finite, and always will be. An open universe is spatially infinite, has always been and always will be.
Current models are homogeneous. If the universe is homogeneous and infinite, then it contains infinitely many photons. If finite, finitely many.
The fact that you can observe photons forever does not automatically mean that there are infinitely many. Their flux is decreasing with time, and you could observe them at a decreasing rate.