I was thinking about writing about the Holocaust, but I am not sure if it's a good idea since I am not sure if people would feel offended if I exaggerate certain events or facts even if it makes the tragedy look more horrible. How can I go about setting my story in the midst of this event without causing too much offense (and assuming I don't want to do a lot of research first)? Are the considerations the same for other historical events?

In general, avoid writing about things you don't know

You are write to have misgivings about writing on a topic you haven't researched. There are a lot of risks involved in doing this. You risk offending people by accident, you risk misrepresenting real people in your work and you risk being called a lazy writer for not doing your research.

Know your topic

With most topics it is nearly impossible to write about it well if you don't know what you are talking about. To write a good story set in London you need to know what London is like. Either you have lived there and know if from experience, or you do extensive research on the culture, layout, transport, architecture and climate. To not do this would open yourself up to poor reviews where readers correct all your mistakes. The same rules apply to historical settings.

Avoid giving details that may be wrong

If you feel that this is the only setting in which your story could work, then you should do the research. If you don't want to then avoid detailing the setting anymore than necessary. It is possible to write a story with a backdrop of a historical setting without giving details of it.

Many stories use the World Wars as a backdrop for the story. The Chronicles of Narnia is set during World War II but doesn't go into detail about the war itself. (Not that I'm suggesting C.S. Lewis didn't do his research) You can do a similar thing if you focus your story on a smaller scale, focus on the characters and the narrative rather than the setting.

Add a disclaimer

This work of fiction is for entertainment purposes only. Any misrepresentation of actual events is unintentional.

This won't stop being being offended, but may prevent the backlash.

I've given some advice on how to get away with writing without research, but my reall message is: Do the research. Your writing will be better for it and you might learn a few things along the way that you would like to include.

Some events are far-off historical events. The most you risk if you write about them without doing the proper research is making a fool of yourself.

Other events are still within living memory. Some of your readers might have lived the event. @SaraCosta says in the comment that not doing research is a sign that "the writer has very little respect for their chosen period / event". Well, there would be living people whom you'd be disrespecting.

One particular way in which you could show disrespect is believing yourself to be exaggerating an event, when in fact you're falling drastically short. I always have a nervous twitch watching Casablanca, when they talk of Laszlo having escaped the Concentration Camps. I think of a concentration camp, I see this (warning graphic):

(source)

Only, of course, when Casablanca was filmed, they didn't know yet. At least, the film-makers didn't.

The issue with believing yourself to be exaggerating an event, when in fact you're falling drastically short, the way you're being disrespectful, is you're underestimating the magnitude of the event. You're underestimating how big it was, how horrifying, how inescapable. You're telling people that their experience is not as big as they know it to have been. That can strike deep.

Also, if an event is still within living memory, it isn't too hard to do the research. You can find accounts and testimonies - written, or on youtube. Because research is easy, it follows that you didn't bother to do the research - you thought it "not worthy of your time", "not sufficiently important". That is how people would see it, and this too they would find offensive.