When writing the conclusion of a thesis, should you say “significantly increased” or just “increased”.?

Depends whether the increase was significant or not.

It depends on the size of the error bars. A significant increase is usually greater than 3 sigma.

Depends if the increase WAS significant. A minor fluctuation would probably not be, a 30% shift would. Depends on the situation. Is it significant?

Depends on how great the increase is, and whether that really is a significant difference. And define what makes it "significant."

That depends. Was the increase significant or was it not?

The use of a qualifier like "significantly" refers to how much of an increase there was. So it goes to the magnitude or importance of that increase. So increases have degrees or levels. It could be an extremely small increase, like 0,00004%. Or it could be a moderate increase, like 15%. Then again, we could have a BIG increase, like 70%. It might have been a really big increase, like 200%. So it all comes down to how much of or how important an increase there was. And that's information we don't have.

You can say significantly if the increase was statistically significant. Otherwise...no...just increased.

It would depend on the intended meaning.