I have been doing this exercise for months now. I started having these problems last week.
What should I do?
Sounds like you need to work out your Brax and possibly lighten the weight.
In my experience when doing barbell curls/concentration curls I would get an intense sharp pain from my brax to my elbow (yours may be to wrist). This is because I transitioned too soon into heavier weights thinking it would help define my biceps oppose to higher volume lighter weights which really would define my biceps.
My sister is a personal trainer and advised I incorporate some hammer curls into my routine. I did an alternating hammer curl style, just 4 sets of 20 with a lighter weight, maybe 25lb DBs...really works your bi's and forearms increasing your muscular endurance and strength overall.
I would firstly do that (incorporate hammer curls into routine), then take the same weight in hammer curls and start there with your barbell curls, or as close as possible maybe a little heavier, see if the soreness goes away, should improve after a week or two of hammer curls.
If the pain is sharp, you may have developed some tendinitis. Sometimes it originates from the elbow and radiates down the forearm, other times it radiates up the biceps. In either case the solution is the same:
An alternate form of curl that is easier on your elbows is the hammer curl. It's worth doing while you are rehabing the elbow. If you were only using the empty bar for curls before, you can load with dumbbells that below that weight.
You will need to increase the pressing work. Particularly because you have both chin-ups and curls in your routine. You should be able to press more than you curl, but keep the reps in the 15-20 range. This has the benefit of moving more blood through the joint and flushing out the inflammation.
If you alternate flat bench, incline bench, and overhead press you will hit everything from different angles. 5 sets of 15-20 reps of any pressing movement a day for 5 days in a week with relatively light weight will be a good place to start with the rehab.
If you are weighting your chinups this is a time to do strict body weight. Additionally, if you can I would switch to pull ups (palms away) while you are rehabbing, and then alternate chin-ups and pull ups afterwards. If you can't do strict pull ups yet, do 4 sets of 8 pull up negatives (just a slow controlled descent). The different orientation will help the weaker muscles in your forearm to catch up and work your grip differently than the barbell curl and chin ups do.