No tourist card is needed for day visits to border towns. People walk across the border to shop by the thousands every day. While a felon might get into a border town easily (usually no check entering a border town but a more thorough check about 20 km south of the border) it's anyone's guess how the US Customs would treat a felon trying to re-enter the US without a passport. I can't believe that reentry would be refused.
From what I see searching whether a felon can get a passport it appears that applications for passports by felons are subject to close scrutiny but not an automatic denial. Drug trafficking convictions would most likely be denied. A passport application would definitely be a good idea - a passport is a form of identification rather than an assurance that the passport holder can go anywhere. Most other countries (especially Canada who routinely denies entrance to any dwi/oui conviction regardless of how long ago) would most likely deny entry by a felon. Mexico border towns, and maybe even the entire country may very well permit entry. It's getting back into the USA that might be a problem.
Sorry about all the bickering. I hope this forum can help folks interested in traveling in Mexico regardless of the means of travel. There's not a lot of other resources on Mexico travel.