ADDED after reading another post: Please NEVER burn off a tick! You cannot save the tick that way, you risk burning your child or yourself, and the tick will feel the heat and its body will release more toxins into the human! This is an "old wives' tale" way to remove a tick -- the tick may come off but you'll have more toxin in your system and can't retain the tick for testing. I've had a couple of advanced first aid courses and we are told never, ever to remove ticks by burning them off or smothering them in Vaseline, which also causes them to release more toxin into the person. Please do not follow this advice! Look online for safe ways to remove a tick with tweezers.
I'd ask the doctor for specifics about symptoms, but Lyme symptoms are very similar to those of many other common illnesses so it can be tough to diagnose. If he seems to come down with a lingering cold, or is just tired a lot, have him tested for Lyme. The bullseye rash is a good indicator but is not universally present with Lyme, so don't go just by the fact there is no bullseye rash yet. It could still appear. It is best if he gets tested because by the time symptoms appear, the Lyme is taking hold and you can knock it out earlier with antibiotics if you start when it's diagnosed rather than when symptoms have been present a while.
It's too bad you flushed the tick. If he ever gets another one, keep it in a plastic ziplock bag and have it checked -- the local health department can actually test the ticks themselves to see if they are carrying Lyme, and if the tick has it, a doctor usually wants to start the person on antibiotics immediately so you don't have to wait for symptoms.
Did you check the rest of his body thoroughly? One tick may mean others. Keep them if you find them! Check his scalp very thoroughly and also check his groin and underarms in particular as well as the rest of him. It's not fun, but ticks like dark and damp places where the skin folds, so the groin is one area they can appear. My friend's sons both had ticks on their scrotums after being outside, so don't hesitate to check your son's scrotum and genitals. Sorry, but those are just the kinds of areas they like.
Also, look up symptoms of Rocky Mountain spotted fever, which ticks also can carry. I know someone who had it as an adult and it was miserable but again -- it acts like other disease so sometimes does not get accurately diagnosed.