Question Should I remove this soft tape before installing the PSU ?

I would expect so.

However, before doing anything, provide more information:

PSU: make, model, wattage, etc. along with a couple more photographs showing PSU and installation case.

My thought being that the foam tapes (there is one on the other end - correct?) are simply for insulation/vibration purposes.

Are the tapes covering the screw holes to be used to secure the PSU to case?

Or just preventing or interferring with the alignment of other screws?
 
I would expect so.

However, before doing anything, provide more information:

PSU: make, model, wattage, etc. along with a couple more photographs showing PSU and installation case.

My thought being that the foam tapes (there is one on the other end - correct?) are simply for insulation/vibration purposes.

Are the tapes covering the screw holes to be used to secure the PSU to case?

Or just preventing or interferring with the alignment of other screws?
The PSU is Antec BP450P with GALAX REV-07 case.

Yes, there is one tape at the other end and that's all of them.

With those tapes, the psu will raise a bit high and can't be properly screwed with the case no matter how hard I push the psu down.

Anyway, I decided to remove the tapes (oops) and now considering your insulation/vibration idea, I think I just messed up.

Here's the PSU perfectly screwed in after tapes removal
View: https://imgur.com/a/mo2zpjr
 
So where the tapes were was on the bottom of the PSU.

And the bottom of the PSU is now sitting (without tapes) on the floor of the case but raised a bit by the bumps originally covered by the tape - correct?

If so, unless there is indeed serious noise and vibration, all should be well.

The tape appears to be just foam and whatever tape was stuck on likely to be the cheapest (price and quality) tape that could be found.

Too thick. But stuck on in lieu of thinner and better quality foam tape. Tape that would have really served the intended purpose.

If noise and vibration become an issue then there may be other ideas and suggestions to help fix the problem(s).
 
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And the bottom of the PSU is now sitting (without tapes) on the floor of the case but raised a bit by the bumps originally covered by the tape - correct?
That's true. I've done some gaming until now and everything is running just fine. Guess those cheap tapes wouldn't be much of a problem then.