I have been having a ton of fun playing Fallout 4 lately [again, building up my Settlements too much and ignoring the Main Storyline!] and ran into a 'small-but-game-stopping' issue where the TAB key appeared to stop working. Hitting TAB wasn't bringing up the Pip-Boy anymore..?
After doing some Troubleshooting and paying attention to what I was doing and pressing, it turned out to be a very simple issue - the TAB key was 'locked' out because of how I had used it in the Console.. Let me explain briefly:
I was playing around with SweetFX in ReShade once again, mainly to take Screenshots as I had lost a ton during a recent Hard Drive failure [I know... but I don't usually back up screenshots or recordings]. To open the Settings/Configuration for ReShade, the hotkey combination is SHIFT+F2. Unfortunately, something in this combination of Fallout/ReShade/Windows 'locks out' the Shift Key after doing this with the Console open in FO4, and when hitting Tilde again to close the Console, this leaves TAB 'disabled' and it no longer brings up the Pip-Boy.
The Fix for this was simple, once I figured it out: hit Tilde again to open up the Command Console, then hit SHIFT once, then hit Tilde again to close the Command Console. That's all there is to it. Now the TAB key worked and I could bring up the all-important-and-totally-cool Pip-Boy once again. This is just a Quick Tip in Text-Mode only then, but I wanted to get it typed and posted so that it could start to help out others that have run into this little problem [that seems to occur when running ReShade in Fallout 4]. HTH