Parts that have shifted during a move are a not uncommon cause of failures that are discovered immediately after moving a computer (whether room-to-room or inter-continental). It's possible that the computer will work after the issue is resolved, however it's also possible any of these items could cause catastrophic damage if the computer is plugged-in to AC mains before being corrected.
Here are some of the more obvious things for which to look after a move and before
connecting the computer to AC mains...
Open the case and verify everything is properly seated and connected. Visual inspection may be insufficient (e.g. a card (videocard, RAM module, etc) may look properly seated but in fact has shifted slightly). If you know what you are doing, consider unplugging and reseating the various components and connectors in the computer.
Also look for loose items in nooks, crannies, and the "floor" of the case.
Moving can even cause a motherboard or other component to shift on the mounting screws and can result in a short-circuit to the case or other component.
Check to make sure the CPU cooling assembly has not detached or shifted from CPU. This will likely
be OK as long as the cooling assembly's mounting lever, cam, and mounting screws appears to be in the proper position (see manual and/or cooling assembly's mounting instructions) and the assembly can't be "wiggled" around. Check this gently. Don't use too much force here. You may also have to check to make sure the assembly's back mounting plate fasteners are all firmly in place and the plate hasn't shifted as that may cause a short-circuit on the backplane of the motherboard.
Look for "missing" screws as well. During a move screws can also be "popped" from their locations and end up bouncing around and ending up shorting something out. I've even found an adapter card bracket's case-mounting screw bouncing around inside a PSU (which I can't confirm came from a move, but seemed likely given the timing of the failure). The card was still in its socket and everything still "looked" OK. A wrong-size threads screw was used and so it wasn't properly held in place. Lesson from that was...unexpected things happen to cause failures.
If external devices (mouse, keyboard, memory stick, etc) were left connected during the move, it's also possible that the wires were "pulled" and caused stress on the connectors to the point where they were damaged. Look for signs of damage, connectors at odd angles, and try disconnecting/re-connecting each device. If it can't be reconnected the socket is likely damaged and may cause power-on problems.
Note: anytime the case is opened, and especially if you are working on any parts, connectors, etc., make sure you follow appropriate personal safety and ESD protocols.