This issue touches (on) personal beliefs, and beliefs always trumps facts.
In this video at 2 minute and 16 second, a speaker used the version with on. But what if I omit the preposition, will it be correct in the context? If not, then why?
I presume it would not be correct since it is a phrasal verb which has a different meaning from the usual
to touch - to move your fingers over something
As a phrasal verb, touch on means to mention something while you are talking or writing.
The conversation was going so well until Marco decided to touch on Diana's death
In the context above, the conversation went from good to bad because Marco mentioned/talked about Diana's death.
The Oxford Dictionary says
touch on (or upon)
1 Deal briefly with (a subject) in written or spoken discussion.
he touches upon several themes from the last chapter
Saying "This issue touches personal beliefs" doesn't really make sense. The sentence needs a slight correction too, to be one of the following:
This issue touches on personal beliefs, and belief always trumps facts.
This issue touches on personal beliefs, and beliefs always trump facts.
NB. The video won't play past 0:46.