I don't know much about this, and I'm a non-native speaker who's never lived in a French-speaking area so I have no personal experience. But, I will try to give a partial answer. I know the question was in French, hunter, but I see that you do understand English as well, and right now I don't feel like I can write this in French (I may edit it later to fix this). Sorry also for the excessive length.
In very broad terms, "ai" tends to represent [ɛ] (aside from at the end of verbs). I think this pronunciation is older. However, there are a number of factors that cause the pronunciation [e] to occur instead for many speakers in many places.
"Approaching Variation in PFC: the Segmental Level", by Noël Nguyen, from Varieties of Spoken French (edited by Sylvain Detey, Jacques Durand, Bernard Laks, and Chantal Lyche), discusses this a bit. Nguyen summarizes Durand and Lyche (2004) as saying the three main factors affecting the pronunciation of mid vowels in non-accented (non-final) syllables are the loi de position, vowel harmony, and root faithfulness (paradigmatic consistency).
Loi de position
The loi de position is the general tendency for vowels to be mid-low in closed syllables (including syllables followed by a word-final schwa) and mid-high in open syllables. When applied consistently, it leads to a complete neutralization of the contrast between mid-low and mid-high vowels such as [ɛ] and [e]. However, in many accents of French, it only applies in some environments.
In general, Southern French accents tend to follow the loi de position more strictly; however, it is unclear to what extent it applies in non-final syllables. It is clear the contrast between mid-low and mid-high vowels is neutralized, but it's less clear what factors govern the actual phone used in non-final syllables. (See comments here Wordreference: Are open-mid and closed-mid vowels still phonemic in French? How many minimal pairs? starting from Nino83, Oct 11, 2015). The two viewpoints seem to be that mid vowels in non-final syllables may still follow the loi de position, or they may have a tendency to be mid-high regardless of syllable structure.
In Northern French accents, front mid vowels are only consistently subject to the loi de position in closed syllables: here only [ɛ] is possible (at least, this is clear for closed final syllables; non-final syllables that are only closed due to schwa syncope, such as the first syllable of médecin, seem to have more variable/uncertain pronunciations). But in open final syllables, mid-high /e/ and mid-low [ɛ] contrast, at least in the standard (which has /ɛ/ in "lait" and /e/ in "clé"). I don't know if there are any accents that maintain a contrast between /ɛ/ and /e/ in final open syllables but have totally neutralized the distinction in non-final open syllables.
I believe the loi de position is even more limited in Belgian and Swiss French accents. "French and Northern Gallo-Romance", by John Charles Smith, in The Oxford Guide to the Romance Languages says that in Swiss French [ɛ] is commonly found in non-final open syllables where standard French has [e], although the example he gives looks odd to me: he says "l'étoile" would be pronounced [lɛtwal] while "les toiles" would be pronounced [letwal], the opposite of what I would have guessed. (Can any user who is more familiar with Swiss French confirm this?)
The following document may also be relevant: La loi de position dans
la phonologie du français contemporain
The next factor mentioned, vowel harmony, tends to affect non-final vowels. It generally shows up as height assimilation, so there is a tendency to use [ɛ] before low or mid-low vowels and [e] before high or mid-high vowels. I don't know the exact regions this is associated with, unfortunately, but I believe it's generally associated with Northern France accents (see "French: A Linguistic Introduction" by Zsuzsanna Fagyal, Douglas Kibbee and Frederic Jenkins).
The last factor, root faithfulness is fairly self-explanatory: there's a tendency to use the same vowel in all forms of a word.
Considering "aimer" specifically: as far as I know, all accents would use [ɛm(ə)] for conjugated forms such as "aime, aimes, aiment" due to the loi de position.
For the infinitive "aimer":
As far as I know, the pronunciation [eme] would definitely be preferred over [ɛme] in Southern French accents due to the neutralization between [ɛ] and [e] and the loi de position.
In Northern French accents, I would expect more variability. For speakers who have /ɛ/ and /e/ as distinct phonemes, the principle of root faithfulness would favor pronouncing "aimer" as /ɛme/. However, Northern French accents also tend to have regressive vowel height assimilation which would favor /eme/. I don't know which of these tendencies wins out in practice. (Note: an interesting example of vowel harmony operating in the opposite direction on this root is "aimable", which is typically said to be pronounced /ɛmabl/ (see for example https://hal.inria.fr/file/index/docid/308395/filename/1560.pdf)
I don't know enough about Swiss and Belgian French to say how the words would be pronounced here. I would guess it is more similar to Northern France than Southern France accents. I found a paper here "The variation of pronunciation in Belgian French", by Philippe Hambye and Anne Catherine Simon, that says the phonemic distinction between /ɛ/ and /e/ "is not stable in non-tonic syllables" for many speakers, especially young ones.
Overall, I doubt the choice of vowel here has much significance. The literature I have read suggests that that /ɛ/ and /e/ are rarely contrastive in non-final syllables in any region, and although speakers vary in their pronunciation, I don't think the pronunciation of the first vowel "aimer" would be a regional shibboleth like the pronunciation of the vowel in "chose".