"I am tired and doing my homework." Is it grammatically correct?

There is nothing ungrammatical about this sentence. It contains a kind of non-sequitur. There are many of these. For example, zeugma involves a kind of non-sequitur. An example of this is as follows.

He walked into the kitchen wrapped in thought and a bath robe.

Here the zeugma involves the same word (wrapped) first metaphorically and then literally. So the sentence jars and, as a result, can be funny.

This one is not exactly zeugma, but something rather like that.

In the case of your example, you can see the grammaticality of it just by substituting but for and.

I am tired but doing my homework.

There is no doubt about the grammar here. But also, there is a logical connection. Being tired is associated with not wanting to work or not being able to work. So if someone says they are tired, we do not expect them to go on and say they are working. The disconnect is a logical rather than a syntactic one. There are even contexts in which there is a perfectly good connection between the two statements in your example.

Mother comes into the daughter's bedroom and says: "Have you thanked Uncle Edward for that lovely present he gave you? It's now two weeks ago, you know." Daughter says "Please, not now, Mother. I'm tired and doing my homework."

Strictly speaking, the title sentence is grammatical, but it sounds unidiomatic because there's no connection between the two predicates. To give some similar examples,

I am raising money and running for president

sounds fine, but

I am lifting weights and running for president

sounds funny. Similarly,

I am tired and thinking in circles

sounds fine — because being tired can make you think in circles — but

I am tired and doing my homework

sounds a little bit funny.

In fact, this is true in general — if we want to connect two completely different things, we generally tend to leave out fewer repeated words than if we're connecting similar things. An example using nouns rather than verbs is:

I inherited her house and garden.

That sentence sounds fine, but you probably want to add an extra her in

I inherited her house and her rhinoceros.

So to make the original sentence idiomatic, you should say

I am tired and am doing my homework.

However, as John Lawler suggests in the comments,

I'm tired and I'm doing my homework

is even more idiomatic, because people generally contract I am.

As with each other answer so far, I’ll confirm that the title sentence is perfectly grammatical. There is no real ambiguity or doubt on that score. In contrast to other answers, though, I find nothing odd, funny-sounding, or unidiomatic about the title sentence. I do not consider it a non sequitur.

In the example sentence, “I am tired and thinking in circles,” being tired directly influences the thinking in circles, because fatigue can cause difficulty thinking through things. This is one way in which clauses can be paired with and that makes sense. But it is not the only way.

In the title sentence, rather than “being tired” causing the doing of homework, it modifies the expressed experience. Doing homework while tired is a worse experience than either separately. It suggests, perhaps, that the author is overworked (or not budgeting their time correctly), because the best thing to do when tired is rest, and the best time to do homework is when you are well-rested. The combination here emphasizes that the speaker is not operating under those ideal circumstances.

It is perfectly reasonable to want to connect these two thoughts with and in order to achieve precisely that understanding. I don’t feel there is any particular need to repeat am here, or I am or I’m. You certainly could, it isn’t wrong to do so, but I don’t think there’s any pressure on the sentence suggesting that the author should do so. I don’t think the repetition or lack thereof significantly affects the understanding or flow of the sentence. Likewise, using but as the conjunction instead of and is definitely an option, but I don’t see it as a superior option. Nor do I see I’m as inherently more natural than I am for this usage. I can easily imagine a native speaker using any or all of these options in just about any permutation.

It’s grammatically acceptable, but a pedantic purist about writing might complain that it isn’t “parallel structure”—one part uses an “-ing” and the other doesn’t.