Mitt Romney's current policy is end all Amtrak subsidies. (a whopping $1 billion/year, or about 1/11 of an aircraft carrier. However, we need those brand-new aircraft carriers to fight against all the aircraft carriers the terrorists have, so whatever.)
What happens if he actually manages that is anyone's guess, but I imagine it would go somewhat like this:
1. Amtrak hikes prices on the long-distance services. Contrary to popular belief, the long-distance trains aren't empty (in fact they're booked solid for months before they leave) but they still lose money because of subsidies keeping the prices low(ish). It's all people on holiday, of course.
2. Amtrak prunes unprofitable routes. (note that this does not necessarily equate with "underused" routes. CSX could hike the price to run trains in the southeast and bam! there goes the Auto-Train.) You'd probably see a lot of mid- to long-distance trains go away. Anything that doesn't make money gets the bin...
3. Amtrak starts to hike prices on the Regionals and the Acela to pay for track maintenance that they are now on the hook for. Infrastructure becomes noticeably worse over time. There simply isn't enough money being made from the Regional, Acela Express, Keystone Service, etc. to keep the Northeast Corridor in tip-top shape. "Temporary" speed restrictions are enacted, trains slow down. Ridership falls, profit falls. People start moving to Megabus and Bolt, even though then you have to hang out at the refugee camp on the dingy sidewalk with 300 other sweaty people in front of 30th Street Station while people try to sell you overpriced water bottles and melted chocolate bars while you wait for the bus. (seriously, Megabus is terrible in Philadelphia.)
4. Amtrak cuts Regional service. Acela service goes the same way after time. Pretty soon the Auto Train is the only thing left because all Amtrak has to pay for is the two depots in Lorton and that other place in that state with all the old people.
5. Norfolk Southern or CSX or some other big nasty angry Class I buys up the NEC and de-electrifies it or makes some very similar short-sighted decision. Meanwhile I-95 becomes a 24/7 parking lot and the airports completely jam with commuter planes. Megabus and Bolt run extra buses to keep up with the demand, but it's not so great with an average speed of 10mph up the now 20-lane New Jersey Turnpike...
6. The entire Northeast collapses because no one can get anywhere anymore. Everyone dies. The end.
That's in the ideal free market Ayn Rand world, though. In reality, this will happen:
1. Mitt Romney proposes to end Amtrak subsidies.
2. Congress says "no" because they want their bimonthly train to East Armpit, State-you-never-heard-of (postal code SN).
3. Amtrak continues to receive meager funding, everyone continues to laugh at it.