No. A dominate effect does not end even if the person leaves for another plane
A Dominate Person effect has a duration of one day per level. It only ends after the spell expires, the target enters a region that negates ongoing magical effects such as a null-magic zone or similar, or the spell if forcibly ended, such as by a dispel effect.
Note that the quote given states "once control is established". The following bit indicates that in order for a command to be issued, the caster and the target must be on the same plane. In other words, the target will continue to perform the last issued command(s) while they are on another plane. It is simply that the caster would not be able to issue new commands to the target while they are on said plane. There is no mention of the effect ending.
The last line of the spell, provided by fectin, also supports this idea with a similar concept:
Protection from evil or a similar spell can prevent you from exercising control or using the telepathic link while the subject is so warded, but such an effect neither prevents the establishment of domination nor dispels it.
Regarding extra-dimensional and non-dimensional effects, if I recall correctly, it has been indicated that no spell effects can cross the boundary of a rope trick or portable hole or bag of holding when they are closed. As such, this should effectively prevent new commands from being issued regardless of whether they are considered on-plane or not, but not otherwise interfere with the Dominate Person effect.
Note that the Transdimensional Power/Spell metafeats indicate that an extradimensional effect is equivalent to a coexistent plane for purposes of the feat, which is about the only indication that they are considered to be a planar effect. Note also that they are still somewhat on the Prime plane, as a dispel effect functions against all known extradimensional/nondimensional effects.
This DM has ruled for personal campaigns that such spaces are considered coexistent bubbles attached to the plane they are on, thus being "off-plane" for purposes related due to the "no effect crosses the boundary" clause which I have generally applied to such spaces, and connected to the plane they are on for purposes such as dispel.