How to make one object gradually disappear as it passes through some object (e.g. teleportation gates)? How to make it gradually appear in other gates?
For a basic effect, you could animate the transparency.
Almost all UI properties are animatable, You can keyframe them by right clicking and selecting Insert Keyframe (1 or I), or hovering over the value pressing I.
You could keyframe the transparency out as your object enters one gate, then keyframe the object location to move it to the other gate (or use a duplicate object) and keyframe the transparency back.
You can do this in Cycles by using a mix node and animating the factor:
This can be done in BI by enabling Transparency in Properties > Materials and animating the Alpha:
Another way you could try is using a Boolean modifier to remove parts of the object as it intersects it. You could then have a duplicate of the object moving out of another boolean cube to make it appear the sections are teleporting: