Effect of zero selection (pressure) on the population health
I need to point out one thing, natural selection does not bring species to perfection.  The best mutant may not be selected for many reasons.

When you have no selection pressure then you have neutral evolution concurring and what takes over instead of natural selection is genetic drift.  Genetic drift is just sample error.  Say you have 1,000 individuals in a population and all of those individuals reproduce, in the next generation all the genotypes would be in the next population.  Now imagine if you only had 10 individuals in the population and only 5 of those individuals were randomly selected to produce 10 individuals in the next generation (the parents die every year).  There is no selection pressure but just by chance some individuals reproduce and some don't.  If you do this over and over again what you will see is that some genes become fixed in the population, meaning that there is no other genotype in the population.

This fixation at small population sizes can occur with either beneficial or deleterious mutations.  Obviously if a mutation is very deleterious the individual will die but slightly deleterious mutations can become fixed in the population as well.

There is an entire field of biology devoted to this study.  It is called population genetics.  The probability for fixation for an allele is its proportion in the population.  This proportion can be calculated by 1/2N, where N is the population size.