Should one avoid using the phrase "note that" in math writings?
"Note that" is a useful rhetorical device in mathematical writing. Its role is to signal to the reader that the statement following it is more noteworthy than other statements, so the reader should, quite literally, take note. While @JeffE sarcastically argues in a comment1 that it must never be used, I disagree; a judicious use of "note that" (in contrast to using it either always as in Jeff's comment, or never) helps to effectively convey nuances of meaning to the reader that are difficult to convey otherwise. This is an example of taking advantage of the English language in mathematical writing. In essence, "note that" punctuates the flow of the text and helps the reader process and make sense of the large amount of information that a typically dense mathematical text often contains, and discern which parts of it are more important and which are less.
Of course, as with any other form of expression, "note that" can be used effectively, or (as JeffE's comment cleverly illustrates) abused horribly.
Now to address your questions:
some incidents lead me to presume that perhaps those readers who are not that mathematically experienced could take "note that" as an imperative or even a condescending sign?
I really can't say whether "note that" will come across as condescending to those who aren't used to its usage in mathematical writing, but I suppose that's possible (and I'd be curious to hear more about those "incidents"). "Note that" is in fact an imperative, so it makes sense that it will be understood as such by anyone, mathematically experienced or not.
if I am writing something to someone that is senior than me in academia and that is not that mathematically experienced [...] would it be suggested that I avoid using "note that"?
My recommendation is that when you are writing to someone who is less mathematically experienced, just like you need to be mindful that you must adapt the mathematical terminology you're using to your reader's expectations and knowledge, it probably makes sense to also adapt the style of writing accordingly to take into account cultural differences across different disciplines. It is probably true that mathematicians are more used to the way "note that" sounds and is typically used, so one needs to tread a bit more carefully with this device when speaking to a non-mathematician. I wouldn't say you should never use it however, just maybe pay a bit more attention to when and how much you use it and give some thought to whether each usage makes sense in the context of how it's likely to come across.
1 JeffE's comment I was referring to, which was moved to chat, said: 'Note that “note that” is simply redundant. Note that omitting it entirely does not change the meaning of the sentence. Note that it’s just a bad writing habit. Note that you don’t need it.'
I'd say 'note that' is supposed to draw attention to something that might otherwise not be noticed but is clear once you think of it. For instance:
Suppose the square root of 2 is the ratio of integers p and q. Note that we may assume that p and q are relatively prime. Then 2q^2=p^2. So p is even. So p^2 mod 4 is 0. So q is also even. This is a contradiction.
If it's not likely to be clear... then write it a different way.
This is a rather subtle and interesting question.
In this math.SE answer I weigh in (mostly negatively) on the usage of "It is easy to see/prove/verify." One key sentence:
To be more specific, I think it is bad writing to say "It is easy to see that X is true" and say no more about how to prove X.
The general principle here is that either it is easy to see X or it isn't. If it really is, just say X. If it isn't, obviously you had better say something. If you're in a gray area -- think very carefully about your audience, but err on the side of explaining too much rather than too little.
There is another aspect to "it is easy to see": it makes a lot more sense to say this than to write it. In a talk, people can only think so quickly, so having someone say "This is not what you should be concentrating on at the moment; don't worry about it for now" can be crucially useful. But this does not have a place in formal mathematical writing, in which the expectation is that the reader is spending some time working through it.
So I don't think "Note that" should be used in place of "It is easy to see that..." However it has other usages.
1) Its literal meaning is "Take note," often with the implication that it will be referred to again. In formal mathematical writing there are other ways to convey this. Writing "Remark: ..." or "Remark X.Y:...." does the same thing and the latter arguably does it better, because you have something specific to cite back to. However "Remark X.Y" is indicative of very formal mathematical writing. In contexts where less formality is assumed / wanted or when writing for people who are not used to the particular format of formal mathematical writing, this might be a bit jarring / off-putting. Moreover, by standard mathematical convention, "Remarks" don't occur in the middle of a proof.
2) It can also just be used purely as filler / transitional words. I agree with @JeffE that the intellectual content of "Note that X" and "X" are identical. Sometimes though you want to insert a few words before X. It is not good to have one piece of mathematical content (especially if it's rich with symbols or other non-ordinary English) coming right up against another, so placing something as separators is a very good idea. So for instance I often write "we observe that...." Maybe this construction is used when the next logical step in the argument is a little less than absolutely immediate, but functionally it is just putting words between steps n and n+1.
I performed the exercise of looking back through various book length lecture notes of mine in honors calculus, number theory and commutative algebra.
In the first two sets of notes, "note that" is rather common: e.g. it occurs nine times in the first 50 pages of the honors calculus notes. Both usages identified above occur. I notice that (as a subcase of 1)) it is often used in a rather conversational/pedagogical way, to give students more help in grabbing onto the more important points in the exposition. This usage is somewhat didactic and could be slightly/subtly off-putting to an audience who does want to be lectured at. Not every usage is great writing: the first is "Note that one subtlety here is that..." Ugh. In fact I would say that maybe 25% of the "note that"'s could be taken out.
The commutative algebra notes are written at the intermediate graduate level and are accordingly a bit more formal: for instance "Remark:..." (though not "Remark X.Y") is used often in them. It is interesting to note [!!] that the "note that"'s are used more sparingly here, and more often in usage 2) than in usage 1).
To come back to the question at hand:
However, some incidents lead me to presume that perhaps those readers who are not that mathematically experienced could take "note that" as an imperative or even a condescending sign? This is much to my surprise. So, if I am writing something to someone that is senior than me in academia and that is not that mathematically experienced (say an applied engineering scholar whose math background is all from the school curriculum), would it be suggested that I avoid using "note that"?
People in different disciplines are going to write differently. Rather than changing all your writing in a way that you think might help, if you are concerned I would suggest just mentioning that you are writing in the style of a mathematician, and to forgive you in advance if it sounds a little weird compared what they're used to. That should more than offset any reader's thoughts that you might be condescending to them.