What is the best thing to do with cabinets that were installed after the tile flooring? Can cabinets be removed and then reinstalled?
Generally you can remove cabinets and re-install them. This of course depends on how they were installed, but usually they are screwed to the wall and to each other. Carefully remove the screws and they will dis-assemble.
As long as someone didn't go crazy with a nail gun they should come out without having to use a sledge hammer - unlike what is so often seen on the "flip that house" shows.
bp21901 is right. Properly installed cabinets should use screws to mount them. Although I have seen some that had about 100 nails AND a few tubes of glue! Nice cabinets may have the screwholes countersunk and then filled to hide the hardware. If the mounting hardware is not visible, use a magnet or stud finder to locate the screws.
Also, be very careful when removing upper cabinets. Even with the doors and contents removed, they can be very heavy.
If the cabinets were installed after the tile, you are actually pretty lucky. There is nothing worse than removing the base cabinets only to find out that they were odd sizes (especially in older homes), and new cabinets won't fit into the footprint of the old cabinets.
For the wall cabinets, I would suggest having several helping hands as well. You might also want to consider building a simple brace to support the cabinets while you are separating them from the wall.