Would a colony with only one male and female collapse?
Populations founded by a single pair are quite common in the lab 

The Drosophila Genetic Reference Panel, one such example of high artificial inbreeding, is a series of inbred lines where the researchers mated brother and sister flies for 20 generations to form the lines, which means they will be highly inbred. When you do these though you often lose many of the initial lines because of inbreeding depression. Inbreeding depression is reduced fitness as a result of increased homozygosity in the lines - deleterious mutations are more frequently expressed. I have tried to find the number of initial pairs they used to make the ~200 DGRP lines but come up unsuccessful, but I've heard it was around 1500 pairs, although that may be well off the mark*. 

What would happen to your mice?

Just like the DGRP lines, there is a high risk that the population would collapse. However, if they do get through a few generations of inbreeding, and are kept in steady conditions (i.e. selection doesn't change once the lines are formed), then they have an ok chance of persisting. If you started with multiple founder pairs then there's a better chance you will still have some (very inbred) mice in a few generations. The success rate will depend on the frequency of deleterious mutations and strength of their effects.

What is inbreeding?

Inbreeding is increased homozygosity in the population than would be expected under random mating. For example, if the frequency of two alleles ($A_1$ and $A_2$) at a gene are $p = 0.1$ and $q = 0.9$ respectively, the expected frequency of $A_1A_1$ homozygotes is 

$p^2 = 0.1^2 = 0.01$ 

the expected frequency of $A_2A_2$ homozygotes is 

$q^2 = 0.9^2 = 0.81$ 

and the expected frequency of heterozygotes is

$A_1A_2$ is $2pq = 2 \times 0.1 \times 0.9 = 0.18$

If the true frequency of heterozygotes ($A_1A_2$) is less than 0.18, then there is inbreeding.

Is there a benefit to inbreeding?

Well inbreeding can be a good thing. Imagine all of those lines that did not collapse. It is possible that a population formed out of those would have higher fitness than the original population, because deleterious mutations may have been purged (lines carrying deleterious mutations are more likely to collapse). So a highly inbred population can prosper. However, that population would also have lower genetic variance. Genetic variance places limits on the potential for adaptation, low genetic variance means low potential to respond to selection. So, if selection is not constant, then inbred populations are more likely to go extinct as they are unable to adapt to novel scenarios.

Inbreeding in nature

One classical example of high inbreeding in nature is in cheetahs, where it is beleived they went through a strong genetic bottleneck ~10,000 years ago. The problems faced by cheetahs today really highlights the problems of high inbreeding because they are struggling to adapt. Genetic bottlenecking is a  serious long-term problem for many species including the Javan rhino, the Kakapo, Mountain Gorillas, and others; even if we can stop poaching and habitat loss wiping out these species in the short-term, they will struggle to adapt to changes in the climate etc..


  "As a species, cheetahs have famously low levels of genetic
  variation. This can probably be attributed to a population bottleneck
  they experienced around 10,000 years ago, barely avoiding extinction
  at the end of the last ice age. However, the situation has worsened in
  modern times. Habitat encroachment and poaching have further reduce
  cheetah numbers, consequently snuffing out even more genetic variation
  and leaving cheetahs even more vulnerable to extinction."




* Supporting material from the DGRP paper would suggest that it was indeed formed from 1500 isofemale lines.