In French, some speakers differentiate between the pronunciation of maître /mɛ:tʁ/ and mettre /mɛtʁ/ - that is, in the first case the /ɛ/ is long and in the second it's short, but that differentiation is not mandatory. Is this an example of free variation?
That is not an example of free variation but loss of an opposition (a phonological merger).
Free variation is a situation where two or more options are available and interchangeable. For example, pronouncing the English word species as /ˈspiːsiːz/ as opposed to /ˈspiːʃiːz/ or vice versa would not convey any difference in meaning, but almost all English speakers are capable of both pronunciations and accept both as grammatical.
In contrast, those who don't differentiate between /mɛːtʁ/ and /mɛtʁ/ are incapable of making (and, in many cases, of perceiving) that distinction, whereas those who do differentiate always make the distinction. In linguists' terms, for the former group of speakers, the phonemic opposition (or contrast) between /ɛː/ and /ɛ/ is said to have been lost, and therefore the phonemes /ɛː/ and /ɛ/ are merged.
I do not agree with Nardog’s answer. The distinction of long and short vowels is a feature of the formal sort of French taught in schools and codified in the prescriptive dictionaries (that is: virtually all French dictionaries). Most forms of spoken French (especially in Paris) have abandoned this distinction, for example pronouncing patte and pâte identically. It is not an issue of not being able to produce this difference. Most speakers will fluctuate between the two styles of speech depending on circumstances. This is not free variation; it is code-shifting.
Nardog is right, but it depends about which variety we are talking.
If we look at ALF: http://cartodialect.imag.fr/cartoDialect/cherche
We can see that the pronounciations used were very heterogeneous. Some pronounced in a similar way, others made differences. Currently, what is it the situation? I am not aware of studies discussing on this topic, then we cannot say for sure that the pronounciations have changed. Maybe, this regional distribution is still present or more homogeneous.
So, effectively, it is not a free variation, but a dialectological variation. Because those individuals don't belong to the same linguistic community. I have read on Wikipedia that the dialectological variation is considered a free variation. This affirmation cannot be defended because that will lead to consider that, in France, one linguistic system exists, there are not varieties. This conception cannot be supported.
It is also possible that there is a sociological variation (in the way of Labov, variationist sociolinguistics), as Fdb raises it. Some, desiring to respect the norm, adapt their pronounciation depending on social circumstances. However, is there a study supporting this hypothesis?