No, teleportation doesn't involve movement unless it says it does
The dimension door spell description states, in part:
You teleport yourself from your current location to any other spot
within range. You arrive at exactly the spot desired. It can be a
place you can see, one you can visualize, or one you can describe by
stating distance and direction, such as "200 feet straight downward"
or "upward to the northwest at a 45- degree angle, 300 feet."
Nowhere in the description of this spell does it say that "the caster casts the spell, then must step through a door that appears", which would use up movement.
Therefore, it depends on what type of teleport they're using.
If they were to use the arcane gate spell (PHB, p. 214), for example, then it would require stepping into the circle:
You create linked teleportation portals that remain open for the
duration. Choose two points on the ground that you can see, one point
within 10 feet of you and one point within 500 feet of you. A circular
portal, 10 feet in diameter, opens over each point. If the portal
would open in the space occupied by a creature, the spell fails, and
the casting is lost.
The portals are two-dimensional glowing rings filled with mist,
hovering inches from the ground and perpendicular to it at the points
you choose. A ring is visible only from one side (your choice), which
is the side that functions as a portal.
Any creature or object entering the portal exits from the other portal
as if the two were adjacent to each other; passing through a portal
from the nonportal side has no effect. The mist that fills each portal
is opaque and blocks vision through it. On your turn, you can rotate
the rings as a bonus action so that the active side faces in a