I feel like rivalries involving the Rangers have declined across the board really. Maybe rivalry hockey isn't as dramatic as it used to be. Even when the Rangers play the Islanders or Devils, it sometimes just feels like another game. Perhaps the decline in physical play over the years is a factor.
A big part is that since the Devils beat the Rangers in the conference finals in 2012, the teams never seem to be good at the same time. There have been no more playoff series between them, and even when more than one has made the playoffs, at least one of them has just been 1st round fodder.
When you look at who made the playoffs between the three in the years since, it becomes pretty clear:
2013 - Rangers & Islanders, Rangers make 2nd round, Islanders lose in 1st round
2014 - Rangers only, lose in SCF
2015 - Rangers & Islanders, Rangers lose in ECF, Islanders lose in 1st round
2016 - Rangers & Islanders, Rangers lose in 1st round, Islanders lose in 2nd round
2017 - Rangers only, lose in 2nd round
2018 - Devils only, lose in 1st round
2019 - Islanders only, lose in 2nd round
2020 - Rangers & Islanders, Rangers lose in qualifying round, Islanders lose in ECF
2021 - Islanders only, lose in ECF
2022 - Rangers only (if standings hold)