Is it ok or mistake to use this words as synonyms? (E.g. to write "we evaluate variance of the estimator by theorem 1" when I mean that we estimate this variance)?
I always thought "evaluate" was used to describe analytical formula results, and "estimate" was used to describe simulation or approximation results.
If you mistakenly use one over the other, people will generally understand what you mean if you are commenting on an online forum or discussing in person. If you are publishing a paper, the distinction will be more important, as papers are often peer-reviewed and academics can be picky. In your case, where you are determining the variance from a sample (not an analytical probability distribution), you would likely use "estimate" since you are not 100% sure of the distribution of the underlying variable, and you are therefore providing an estimate of the true variance based on the sample data you collected.
In conversation, the two are generally interchangeable, in my experience.