During teething I'd use tylenol &/or ibuprofen as needed. Happy little guy during the day, but needed it to sleep. Typically a few days to a few weeks. Which was absolutely fine with all the med pros in his life (his peds, doctors in the fam, nurses in the fam, pharmacists in the fam)... because he had absolutely no history that would contraindicate. ((For example, a baby with stomach problems you tend to use tylenol instead of ibuprofen)).
ANYTHING when given too much of can cause scary sounding side effects. Vitamin A causes blindness & liver failure, Water causes water toxicity, sugar can shut down the islelets of langerhans (part of the pancreas that sends regulates/produces insulin/blood sugar), calcium causes kidney stones and seizures... ANYTHING given too much of one can OD and cause major problems. But you don't restrict water, vitA, sugar, calcium. You just stay careful not to overdo it. AKA balance. In meds... that means staying withing the dosing limits per weight (which are NOT the maxium that can be given... it's just the *absolutely safe* amounts that can be given... for high or stubborn fevers, for example, call the 24/7 nurse line to get the *actual* dose per weight.
You can give (or take) the standard dose of either tylenol or ibuprofen every single day for YEARS and have absolutely no side effects from them. In fact, most people don't. The reason why "after 3 day call your doctor" is because MOST things resolve within 3 days, and if they don't it's a sign of something a doctor needs to look at.
I've had to do pain management for YEARS. Because I was taking 800mg of Ibuprofen every 4 hours (to deal with massive swelling) I'd get my LIVER function tested (even though it's more common to cause stomach problems) every 6mo while in middle school. I was eventually able to back off of the ibuprofen.
If there's no underlying disease (liver, stomach, etc.), allergy, or illness... it is AS SAFE to take tylenol or ibuprofen once or twice a day at the max dose on the bottle, as it is to drink water, or eat carrots. Our bodies deal with them quite nicely. Not addicitve, filtered easily. Teething lasts longer than most things (excluding chronic pain) but it only lasts a few months, and it's on and off. Week here week or two there. Don't feel bad about giving your baby necessary medicine.
I've CUT teeth as an adult. It hurts as much as getting a tooth pulled without novacain (also had that happen in the back of beyond). Tylenol just takes the unbearable edge off of it. If you would take a pain reliever when you're in pain, I don't understand why parents just let their kids stay in pain.
Again, as always, (since I'm just a faceless typer on the internet) don't take my word for it. Check with your ped or pharmacist.