As the adverb directly relating to the adjective 많다 (to be much/many), 많이 itself means in a large number/amount. Because 너무 (etymologically including the verb, 넘다 (to exceed)) is an adverb meaning excessively, or rather exceedingly, 너무 surpasses 많이 in meaning; you can leave out 많이 of 너무 많이 (but it does not mean that 너무 많이 can always be used instead of 너무). The following sentences can also be used without changing the meanings that your sentences have:
- 손님, 너무 많이 걱정하지 마십시오.
- 손님, 걱정 너무 많이 하지 마십시오.
- 너무 운동하지 마십시오.
- 운동 너무 하지 마십시오.
Now you might think that the phrase, 너무 많이, seems unnecessary, but let's look at another phrase, 너무 많다.
- 나는 걱정이 너무 많다. (I have too many things to worry about.)
- 나는 오늘 할 운동이 너무 많다. (I need to do an excessive amount of exercise today.)
As I have mentioned above, 많다 is an adjective. Normal Korean sentences cannot end with an adverb such as 너무 but can end with an adjective attached to a sentence ending, so you cannot omit 많다 from these example sentences. When you change 많다 to 많이 and rewrite part of them, the following sentences are constructed:
- 나는 걱정을 너무 많이 한다. (I worry too much.)
- 나는 오늘 운동을 너무 많이 했다. (I worked out too much today.)
For these sentences, it is the speaker's choice to leave out or hold 많이.
When the word modified by 너무 is incompatible with the idea of a large number/amount, 너무 많이 cannot be used instead of 너무. Examples are the following:
내 말을 너무 그대로 받아들이지 마라. (Don't take me too literally.)
나는 시간이 너무 없다. (I am too pressed for time.)
양이 너무 적다. (The amount is too small.)
Note: As LegenDUST mentioned, 너무 하다 (너무 한다, 너무 했다, ...) and 너무하다 (너무한다, 너무했다, ...) are different.
Note: As for 너무 조금 (조금 is the antonym of 많이), you cannot omit 조금.