What will happen to liquid nitrogen in a storage vessel if the pressure is increased beyond its critical point pressure (33.99 bar abs)? Will it remain as liquid?
Yes. You can place as much pressure as you like on a liquid. Above the critical temperature there is no difference between liquid and vapor phase so, basically, the phases do not exist.
If you're asking what will happen if you decide to pour out some of the nitrogen, I would think that it would immediately vaporize upon hitting normal pressure, since it's well above its boiling point.
Assuming the initial temperature is -320F and pressure is atmospheric (14.69 psia), it will just remain a liquid until the critical temperature is reached due to heat transfer in the vessel. Then once you reach the critical temperature and stay above the critical pressure, it will be a supercritical fluid.
Then things get wierd. Pipe transport will probably deliver a volume equivalent to what you would get if it was a liguid, but at the pressure drop you would expect for a gas.