If it were a tear you would definately have some limitations in function and from what you've said you just have pain with specific exercises. I doubt it is a bursitis. More than likely it's a tendonitis of the biceps tendon. Don't stop working out, just avoid what-ever causes extreme pain and find alternatives. Instead of flat bench, try decline or lockouts instead. Warm up the area very well and try doing some pendulums and band internal and external rotations before lifting. You want to keep that area moving to help facilitate healing, just avoid the painful motions.
Also, try some anti-inflammatories to help out as well. Check your rear delts as well and make sure there are no imbalances. Look at your shoulder flexibility as well and try to improve upon that. That area of your shoulder doesn't have much room and any inflammation can easily cause some impindgement.