I am a Doctor of Chiropractic (D.C.) who's pratice specializes in sports injury. I have gone through this same type of weight lifting problem myself when I was in my early 20's. So here is my opinion for elbow tendonitis....
The solution is not taking some kind of anti inflammatory drugs that will damage your liver and kidneys over time, the solution to this type of condition is the correction of the biomechanics used with the lift (curl). Using a straight bar to do curls with is damaging to your elbow joints from the very start. When you grab a straight bar in a bicep curl position, you are externally rotating your elbow joint (turning the palms of your hands out which turns elbow out) to its limit and then curling the weight. This amount of un-natural positioning of the elbow causes tremendous amounts of wear and tear on the elbow joint (ligaments, tendons, cartilage). TO CORRECT THIS, due your curls not using a bar but instead using dumbells or universal grips with a more neutral type of position. Make sure your arms are not turned out too far; about 45 degrees from your side at the most. Anotherwords, when you are standing up with your arms naturally at your side, look at the way your hands are positioned. The palms of your hands are facing the sides of your legs. Now turn your hands out about 45 degrees (1/4 turn) and this is the position you should be holding the weight when curling. The position should look like its between a hammer curl position and a straight bar position.
If you presently have elbow and/or forearm pain, make sure you use very light weight if you are to curl. There should be extremely little if no pain at all when curling with the correct amount of weight. Over the next couple months increase the amount of weight gradually with the new curling position I explained above. Make sure that the curls do not bother your elbows/forearms. If they do, decrease the weight curled until it doesn't cause pain. It will take time 4+ weeks on average to start getting back at the weight you want to curl. REMENBER, ICE IS YOUR FRIEND, I RECOMMEND YOU ICE YOUR PAIN AREA ABOUT 4-5 TIMES DAILY WITH AT LEAST ONE HOUR IN BETWEEN ICINGS. Ice for about 25 minutes each time with one layer of t-shirt in between the ice pack and your skin. Becareful of ice burns.
Also doing very light forearm curls and extensions with very light dumbells is a good idea to strenghten the damaged tissue in the elbow joint.
Treatment from a Chiropractor who deals with extremity (elbow, etc) and spinal adjusting may help greatly.
What I wrote above is my opinion as a Doctor. Treatment plans will vary from patient to patient depending on the exact condition and the severity of it. If you decide to take any of my opinions, do at your own risk.
If you have any questions, post them in this forum and I will give you my opinion!