Are there some Torah opinions that say one should not move oneself to tears on Rosh Hashana, when praying?
If so, why?
Yes, this is the opinion of the Vilna Gaon [Maaseh Rav 207]. It is based on the pasuk in Nechemia 8:9 that Rosh Hashana is intended to be a day of rejoicing in the “coronation” of Hashem as our KIng.
Some sources explain this refers to tears of depression and sadness, but tears from feeling closeness to Hashem is permitted and even ideal.