If a person finds a tick on their body or the body of a small child on shabbos may the tick be removed? Is there any problem with trapping, the tick being muktzah, or creating a wound by removing it or does the possibility of contracting a dangerous disease (Lyme poisoning) outweigh other factors?
If one has a flea on his skin and is biting him, he may remove it, but should not kill it. There is no violation of the melacha of tzad because these items are not generally hunted and are prohibited only Rabbinically, which is waived since there is pain. (Mishne Berurah 316:36,37)
A tick bite in addition to the above heter also involves potential personal danger from Lyme disease and is certainly permitted (even if the tick is not actively biting him) since the danger is viewed as a situation of tzaar (pain) and possibly pikuach nefesh and any Rabbinic prohibitions are certainly waived (Orach Chaim 316:10).
This summary of the laws comes close to directly answering the question, but it leaves out a direct response to a case of an insect (do not kill or trap, since they pose no true threat but merely hurt) that actually can cause serious harm or even death (whereas an "animal" in such a case, such as a rabid dog should be killed).
The article, quite notably, points out that, if someone were deathly allergic to bee stings, one should endeavor to chase away the bee rather than trap or kill it. I don't believe this advice/Psak is universally followed. I've often seen/heard consideration given to whether a small child is around, for which there is generally given leniency. But, I think, though I'm no Posek, if a Rav were pushed, he'd rule that if the bee really does pose a threat, one can kill it. (The article does note that you can trap an insect that isn't normally hunted, though, as that is only a rabbinic prohibition - as opposed to a large animal that is normally hunted, which is a biblical prohibition.)
I was once in the home of a Rav of mine from high school, and he mentioned that where he lived there were scorpions that were considered to be so dangerous that the Rav there ruled that if a person saw one across the street on Shabbath, he should run across the street and kill it.
I don't know, as I'm not a Posek, but I think there is enough gray area to be concerned about Safek Nefashoth and remove the tick.
One caveat, though, is that if Shabbath is nearly over, it would probably be best to wait.
If you're talking about a child, and the child was playing in a pile of leaves or grass clippings Wednesday and hadn't bathed until just prior to Shabbath so the tick wasn't noticed, and by the time you get the child out of the bath and dried off and dressed it is time to light candles, I'd think it would be prudent to remove the tick, even though you may end up killing the tick in the process after Shabbath has set in.