No, it won’t.
Other countries have eliminated their penny equivalent. Nothing at all changes. Prices don’t have to be rounded to nickels. You will still write checks and make electronic payments to the penny. Credit card charges will still be to the penny. It will only affect actual cash payments. And even then as long as pennies are still around, they will likely be accepted. Just don’t expect stores to have them available to make change. Most will just round to the nearest nickle and there won’t be any massive shift of money to anyone. We all know how to round. (Or at least those with an actual 8th grade education will - I am convinced that a sizable fraction** of the population does not operate at the 8th grade level.)
–Peter
** Fractions - another thing that separates those with 8th grade functions from those without.