Physically in the NES you cannot press Left and Right or Up and Down at the same time.
Is it allowed for speedruns to activate this option in the emulator?
To clarify, Left+Right and Up+Down is usually banned BECAUSE it's impossible on NES. So allowing it on emulators would give an unfair advantage to emu runners.
So, it is best to show the input of the emulator on the screen to check that UP + DOWN / RIGHT + LEFT is not done? (I say this because I usually disable it)
A lot of NES Speedrun boards by now link to this document as a general guideline: https://docs.google.com/document/d/1apP69VDIzUx-iVO_sj5YjTxBAE2ukHPa6kl33AOW9hA/edit
This should apply to basically all NES speedruns (and most of the stuff is also a good base line for other consoles)