One grammar books tells me that in some cases the marker of infinitives “to” should be omitted, and another grammar book tells me that in these cases the marker of infinitives “to” can be omitted. I noticed that there is a difference between these two statements, which is between “should be” and “can be”. So I am wondering which of the two sentences in each following pair is wrong? Why? If both cases are acceptable, what are the differences?
1. Did you hear him go out?

Did you hear him to go out?
2. I would have him wait for me.

I would have him to wait for me.
3. Last night I did nothing but watch TV.

Last night I did nothing but to watch TV.
4. All we can do is go at once.

All we can do is to go at once.
5. I can’t but tell her the truth.

I can’t but to tell her the truth.
6. We had better do it at once.

We had better to do it at once.
7. Why not go out for a walk?

Why not to go out for walk?
8. Why do it that way?

Why to do it that way?
9. I must make do with the old suit.

I must make to do with the old suit.
10. The old man would rather drink tea than drink coffee.

The old man would rather to drink tea that to drink coffee.
Thank you very much!
You won't go wrong by choosing the first of each pair.
The second in example 4 is also possible. In all other cases the second of the two is not a very good choice.

I mostly agree with your first grammar book. To my ear all should be omitted. Only #4 sounds acceptable to me both ways, and #5 is a maybe.

I think #9 is a fixed idiom "to make do."

I looked for synonyms which would definitely require the "to" and only found one: #10 - "would prefer to drink tea"

(I hope I haven't taken this out of action. I'd like to think about it some more.)
Students: We have free audio pronunciation exercises.
 CalifJim's reply was promoted to an answer.
Thank you very much!

So the bottom line of language use is to be free of misunderstanding, and then try to follow the conventions of native speakers as much as possible.