Will removing the insulation from an unfinished basement cause it to be a lot colder?
Good question. Under the circumstances, I feel you would be better off leaving the existing insulation alone, or simply cover it with a more attractive plastic sheet or some inexpensive fabric. perhaps a curtain type wall over the insulation?  If you remove it without a plan to replace the insulation, the basement will be somewhat cooler. There are several ways to insulate the walls without framing, but adhering rigid foam insulation is expensive, up to $80 per 4X8 sheet, and spray on foam is more expensive than that and should be professionally applied.