"You're doing it wrong."
(Actually it's possible your pain is being caused by something other than doing it wrong. But doing it wrong may be the most common cause of this.)
There are multiple possible medical causes for any one symptom like this. A better way to determine the source of your pain is to get it checked out in person by a sports medicine doctor or an orthopedist specializing in upper extremities. These docs have way more training and experience diagnosing problems like this than anybody here. They can also get your answers to their questions and can gently manipulate your arm in different ways to see what causes your pain to increase. All this can help them narrow this down quickly to the right diagnosis. A diagnostic image may help too.
But when pain is worse during and immediately after exercise, and then clears up with rest, and then comes back when the exercise resumes, it's *usually* an overuse injury like tendinitis. You may be doing the exercise in a way that stresses the tendon (or whatever is causing the pain), or you may not be resting long enough--common issues with driven, self-taught weight-lifters or extreme fitness folks. A session with a personal trainer or physical therapist could reveal if they think you're "doing it wrong."
But you could have a different non-tendon issue like a bone chip from the elbow tip. Then pain relief from resting could be temporary and come right back with exercise, and "doing it right" may not help.
Here's an example of why having a doctor there to get your answers to their questions is very important in diagnosing this: "outside of the forearm" doesn't tell a doctor where exactly the pain is. If you said that to a doctor, they'd say, "Point to where it hurts." (Why? Look down at your forearm. Now rotate your hand 180 degrees to the opposite side. The outside of your forearm is now the inside of your forearm. MDs use different medical terminology--anterior. posterior, lateral, or medial--to designate a forearm side.)
It could be tennis elbow (a specific type of tendinitis that, as you realize, can be caused by many things other than playing a lot of tennis). Here's very reliable info on that issue:
http://orthoinfo.aaos.org/topic.cfm?topic=a00068
Notice they say the pain will be felt on the "outside of the elbow." That does specify one location: the bony part of your elbow, where you can rest a hand without interference while flexing your biceps. If your forearm pain starts at the "outside" of the elbow and radiates towards the hand, yes, it could be tennis elbow. Tennis elbow that's been going on (or off and on) for a while is often very slow to heal and may benefit from therapeutic exercise guided by a physical therapist. But it's possible it's something else.
With many overuse injuries, you don't need to know the specific tendon or other body part that's acting up to get relief. What you may need to do is RICE:
http://fairmountsoccer.org/rice-injury-first-aid/
*especially the resting part* for *at least a week after* all pain is *gone*, and then resume the exercise more gradually than you've been resuming up to now.
If you've really given RICE a good shot, then you should get it checked by a sports medicine doc or an orthopedist. Or consider seeing a trainer and asking, "Am I doing this exercise incorrectly in a way that can cause pain?", and again, if that doesn't help, go see a doctor. You may be better off seeing a specialist than a family doctor though.
Wishing you good luck, good care, and pain relief soon.