No Stupid Questions

My opinion is don't do it. Just be yourself. For many situations, you're only going to see the person for a couple of minutes, at most, so who cares what they think? If the conversation goes beyond that, you might find you want to make a connection, and now you've started the whole thing with a lie. Not a good look, and also you won't feel good about it.

My perspective is as a white guy who travels extensively (almost 40 countries now, 6 continents).

As others have mentioned, people don't judge you immediately based on where you come from. In my experience, they will most likely already have judged you based on your looks. The reaction will entirely depend on where you go, and to a smaller extent how you act.

For example, in (East) Africa, you probably won't get treated as nicely as white people do, from what I've heard. And white people get stared and pointed at a lot (unless you're in the full immigrant/expat part of a big city), harassed for money, and called mzuungu constantly. Generally this is all "in a nice way" for white people (but I got tired of it and didn't like it), but talking to other travellers, if you're "Asian", it isn't so "nice."

In southeast Asia it will be a mixed bag depending on what you look like. You'll probably fly under the radar, and have to deal more with people being shocked at your lack of speaking the local language. I saw this happen to Asian-American people a few times.

In Central and South America, excluding Peru and Brazil, be prepared for outright racism. Peru and Brazil have relatively large Chinese and Japanese descendant populations, but in the rest, I believe there is a lot of anti-Asian prejudice. They'll call you "chino", make eye/hand gestures, etc. Not all of this is in a hostile way, but I imagine it can get tiring or worse. Some will outright ignore you in stores and restaurants or even give you scowls. I saw all of this happen to other travellers, and heard even more stories.

I don't know if I can talk about Europe or Australia. I don't stand out there and I haven't really talked to anyone who does. I imagine it really varies by country and maybe even neighbourhood.

One thing to remember is that in places where English is a second language, they can't tell where you're from based on accent as much. You could say you're from Australia with your American accent and they'll full on believe you.

Oh and another one, since you were talking about using Mandarin or Cantonese with an accent... There are very few places outside of China, Hong Kong, Taiwan where people will speak any Chinese, let alone recognize the accent. Maybe Malaysia and Singapore? English is the de facto language of travel pretty much everywhere. Acting like you don't know it won't get you anywhere. You'd have to do a fake Chinese accent while talking English. Practice your best Uncle Roger I guess.

Final Bonus Pro Tip: the best thing you can do for yourself when travelling to countries that aren't primarily English, is learn how to say hello and thank you in the local language. If you initiate a conversation (e.g. respond to a stare) with a local language "hello", you will get a smile and hello back and tension will ease. The rarer the language, the bigger the smile.