Does macOS High Sierra 10.13's EFI firmware check cause problems for MacPro3,1 2008 installations?
Short answer: No, there's nothing you need to worry about eficheck. 

First of: eficheck is not a blocker, at least in its current state. Also I wouldn't call it an annoyance as it's supposed to be a security feature on supported system and the prompt it generates only appears once and remembers your decision whether or not you want to send data about your EFI to Apple.

Coming back to your question, eficheck shouldn't affect your system in anyway. I'm currently running High Sierra on my MacBook5,1 patched with the High Sierra Patcher Tool and it runs fine. Executing eficheck manually from the Terminal just throws the following error:

ReadBinaryFromKernel: No matching services found. Either this system is not supported by eficheck, or you need to re-load the kext
IntegrityCheck: couldn't get EFI contents from kext


Meaning the tool cannot read the EFI anyway. 

I guess the same would apply to your MacPro3,1, I'm not entirely sure though. But even assuming it does, it wouldn't block anything and I doubt it would prompt you the message because your EFI firmware likely isn't modified anyway. And even if eficheck gets triggered for some reason: you could just click the prompt away once and it never comes back again.