Other British colonies certainly took notice of the situation, and the Continental Congress even intended to invite representatives from other colonies including Québec, West and East Florida, Nova Scotia, and St. John's Island. Only Québec actually received letters of invitation, and they declined to attend.
The reasons why more colonies didn't jump on the independence bandwagon obviously vary in each case, but here are some notable ones. Québec opted not to join because it was tired of war and generally content with the state of affairs. Québec received special legal privileges under the Québec Act of 1774, and overall the colonists there weren't angry enough to warrant revolution.
I'll be blunt and say that I'm not too familiar with the political situation in Britain's colonies in India and the Caribbean at the time. However, both of these regions were far more valuable to the British economically. Caribbean sugar plantations and India's vast human and natural resources were essential the the financial wellbeing of the Empire, and while American taxes and timber were certainly nice they weren't enough to make or break the British economically. Consequentially, the British were far more invested in their Indian possessions (which were threatened by French and French-backed Indian opposition) and the Caribbean. Rebellion in these colonies would have been much more difficult than it was for the Thirteen Colonies.
It's possible I'm missing a conflict that took place in India or elsewhere beteween 1765 and 1783, as I said I'm especially lacking in my knowledge of colonial Indian history. But barring that, there were no major rebellions that coincided with the American Revolutionary War.
If you're interested in learning more about the Canadian perspective, there's an excellent answer from a user on another forum that I'll link below.
^ Look for Dale's response, it should stick out.
There were several conflicts in India during this time. Closest in time to the American war were the First Anglo-Maratha War (1775-82) and Second Anglo-Mysore War (1780-84). However, these weren't really a matter of Indians hearing that America was rebelling and taking advantage of it. They were rather a matter of the local British (mostly the East India Company) fighting to gain more land, and being backed up by British and Hanoverian troops (as Britain and Hanover at that time were in personal union). At this time, Britain didn't have nearly as many colonies as it would later, and was still in its expansionist stage, trying to increase its small but lucrative possessions in India and the Caribbean. You can see their territory at in 1763 in this map: http://cn.bing.com/images/search?q=british+empire+1800+map&view=detailv2&&id=E2266562B4EA067417B5F6CAAA438D64E5D2765F&selectedIndex=3&ccid=uyte8zne&simid=608015886656668498&thid=OIP.Mbb2b5ef339de4e8ca9d97314d60e5408o0&ajaxhist=0
That's a few years before this, but they hadn't gained a huge amount since then (and they'd lost the Phillipines).
There were of course the Caribbean colonies, but these had small populations and were extremely dependent on trade, making independence less attractive. During the war, thousands of British citizens (and slaves) died due to French blockades of the Jamaicas, which shows how dependent they were on foreign trade. They were also incredibly important to the British economy: Jamaica was worth more than all the 13 colonies. So they were much easier for the British to control, and much more important for them to try to control.
So it was mainly the French who tried to take advantage of Britain's fighting with America, joined by the Spanish and later the Dutch. In fact, one community (the Nairs) under the rule of Mysore (a French ally) took advantage of the war to rebel, with British help, though the French helped Mysore to put the rebellion down.