I have been on coumadin for a couple of years, and I also take a daily 81mg enteric coated aspirin. My doctors are fully aware of this regimen, as they prescribed it.
Aspirin and coumadin both do "thin the blood" (note this is NOT literally what they do), but in completely different ways. Coumadin interferes with vitamin K-dependent blood clotting factors. Aspirin blocks the production of thromboxane A-2 by blood platelets, so their ability to clump and form a clot is reduced.
Aspirin has no effect on the INR value (that is, INR is not a method for measuring the anti-clotting affects of aspirin, since aspirin's action is not detected in the INR measurement).
However, the overall risk of bleeding is raised when on both, then on either by themselves. That's why I take the enteric coated aspirin, to avoid irritating my stomach lining and risking bleeding from ulcerations. It's also best to have your INR checked ever 4-8 weeks (I get mine done monthly), since coumadin's affects are influenced by so many variables (eg. you should be avoiding foods high in vit. K, like broccoli, dark green leafy veggies, green tea).
A few months ago my INR (target range of 2.5-3.0) shot up to 4.7 - why? I have no idea, and neither do my doctors. We've since found that it now sits nicely at about 3.0 with a slight reduction in my total weekly coumadin dose (we changed it so that the cumalitive weekly dose is down by 5mg). It is not unusual for INR to change periodically - that's why regular INR blood tests must be done to monitor your coumadin dose. If INR is high, the standard thing to do is skip a dose or two, then go to a reduced daily dose, retest the INR in a week or two, and go from there.
Being on coumadin means constant monitoring (every 4-8 weeks) - I cannot emphasize that enough. It is a very safe and effective drug for controlling risks of clot-related problems, but it is also easily affected by diet. Regular monitoring is necessary for safe use.