Both Motrin (ibuprofen) and Aleve (naproxen sodium) have mentions on their labels that the product may cause stomach bleeding.
Motrin labels warnings more thoroughly, as the warning about possible stomach bleeding is placed in the general warnings, and by pointing out that you should see a doctor before taking it if you are on anticoagulants. Other ibuprofen products include Advil, Medipren, Nuprin, Act-3, Brufen, Nurofen and many store-brand names.
For some reason, Aleve places its warning about possible stomach bleeding in the alcohol warning section. It doesn't mention anticoagulants specifically, only a general mention of any other drugs taken regularly. Naproxen sodium is also available as Naprosyn or in store-name brands, as it is also open to generic manufacturers.
Another NSAID OTC pain reliever that I favor for backaches, Orudis (ketoprofen) has warnings about not using with Coumadin.
These products do not affect INR or platelet clotting, but do have some risk for stomach bleeding, which can become serious for Coumadin/warfarin users. Likely, most folks on ACT (AntiCoagulant Therapy) will get away fine with occasional use of these products as suggested by their doctors. However, it's much less likely that they would get away with continuing use of these products. The problem is, you have no way of knowing if you are "most folks." So, bypassing these options altogether is a safer path, if you are on ACT (Coumadin or warfarin).
Acetaminophen (Tylenol and many other brands and store brands) carries little risk of stomach irritation, and tends to be less interactive with other drugs. However, overdosing can happen easier and can be quite serious with acetaminophen products, so follow the label instructions. If your doctor has you taking higher doses, follow your doctor's instructions.
It always seems to me that there should be other painkillers besides opioids and acetaminophen that will work for ACT patients.
Best wishes,