The short answer would be no
Two main reasons for this. First, the phonetical distinction between e and ѣ was lost de-facto about two centuries before the letter was officially abolished. Second, for historical reasons modern Russian is one of the most homogenous languages in the world, the number of dialectal variances is quite low.
The slightly longer answer would be - there's some evidence that in some semi-moribund dialects there were indeed some traces of e/ ѣ distinction.
Here's a quote:
В подобных диалектах фонемы верхне-среднего подъема /ѣ/ и /ω/ могут
реализоваться дифтонгами [ие] и [уо], у которых изменение тембра
происходит за счет понижения подъема гласного: л[ие]с, к[уо]т или
монофтонгами верхне-среднего подъема. Эти диалектные фонемы, как и их
звуковые реализации, называются е закрытым и о закрытым. Они
противопоставлены фонемам средне-нижнего подъема /е/ и /о/,
реализациями которых в говорах бывают монофтонги средне-нижнего
подъема либо дифтонги типа [еи], [оу];
So basically, it is claimed that in some dialects there exist so called е закрытое (closed "e") and in some о закрытое (closed "o"). As a sidenote, there are dialects with both o closed and e closed, but no dialects with only o closed.
But you should take anything you here about phonetic differences in Russian dialects with a grain of salt. Even relatively new sources, dated back to 80s or 90s can easily be obsolete - the dialectal landscape changed drastically since then.