"I need advice on whether or not I can refinish the top, or if it's possible to lift the veneer off the tabletop, and just tile the entire top?"

The answer to first question is no,or more specifically you can refinish it but the swelling and cracking will still be present. The answer to second question is yes,that's the best option if you have the skill and proper tools. The answer to question not asked is,some people remove damaged veneer and install new veneer in it's place,require's more skill than tileing option.

Thanks so much, this is very helpful!