Pressure not increasing as corny keg warms

So, 3 psi isn't much. I would expect that the lid isn't seated correctly if you only used 3 psi. Usually takes 30 psi to fully seat a lid.

 

Posts and other connections with o-rings or gaskets can sometimes be torqued too much; squeezing, squishing and deforming them to a point they don't make a good seal.
Have you checked the connections and lid for leaks? If not, spray some Starsan on all connections and around the lid, watching for bubbles.
Leaks aren't usually too difficult to locate but can sometimes be a challenge to stop. More oomph isn't always the remedy.

 

To follow up on what @jbakajust1 said about seating the lid, you don't need to keep 30 psi on it for hours or even minutes. A few seconds should be enough to seat the lid, then bleed off the excess pressure via your PRV (if equipped) or the spunding valve, or by depressing the gas post poppet.

ETA: This has worked for me.
Unlock the lid as if you were going to remove the lid, but don't remove it. Drop it just a tad then pull up firmly as if you were going to close it up and put the CO2 to it. As pressure increases, lock the lid. I'd do that in case the gasket somehow got cattywhompus around the lid.

 

The accuracy of your pressure gauge is questionable. They all are. Below around 8 or 9 psi things get wonky. At 3 psi it's a ral crap shoot. It's a $5 piece of equipment after all.
I would not sweat it too much.

Cheers

 

A day at 3 PSI isn't going to get much CO2 dissolved, and it's nowhere near equilibrium yet, so once you remove the gas, the headspace CO2 will mostly go into solution, bringing your headspace pressure down to at/near 0 gauge.

That's not to say that you might not also have a leak, but a leak isn't required to cause what you're seeing.

 

If you are saying you were dispensing beer without the keg being on gas, then yes, the CO2 level in the beer will go (already has gone) down, because it seeks a new equilibrium between the now smaller liquid volume and larger headspace volume.

If you are saying that you were dispensing beer with the keg being on gas, and if the PSI at the time was appropriate to maintain the CO2 volumes at the beer's temperature, then you could take it off the gas afterwards, and assuming no leaks, your CO2 volumes in the beer will not have changed.

But in general, the best way to keep your beer at the carbonation level you want is to put it on gas at the PSI required to reach and maintain your target CO2 volumes at the beers temperature, and just keep it there. i.e. don't screw around with fluctuating temperatures and/or turning the gas on and off. There are lots of charts online to tell you what that PSI would be.