As a native speaker of American English, I would argue that "over" here is acting as a Spatial Deixis:
Spatial deixis, or place deixis, concerns itself with the spatial
locations relevant to an utterance. ... [T]he locations may be either
those of the speaker and addressee or those of persons or objects
being referred to. The most salient English examples are the adverbs
here and there... although those are far from exclusive.
So "here" and "there" are well-known spatial deixes in English. To simplify, If something is close, it is "here", and if it's further away, it's "there". However, less well-recognized are the compound deixes, such as "over there". So if something is far away, further than "there", it can be said to be "over there".
Let's look at Spanish spatial deixes, and compare them to English:
In English, you have two distances: here and there. In Spanish we have three: here, there and over there.
Here: Aquí, Acá
Over there: Allí, Allá
Translators use "over there" where Spanish uses "allí" "allá", which means it functions as a spatial deixis.
So, to answer the question at hand: "Get over to the dining room" means, to my ear, that the dining room is not "here"-- not close-- nor "there"-- not the next room over-- but "over there"-- that to get to the dining room, you would have to traverse some distance.
Here is a way you can test my interpretation, at least in the example referenced from School Swap: When the students are admonished to go from their dorm room to the dining hall, is the dining hall nearby? Or do they have to go through several rooms, perhaps down a hallway or staircase, or even to another building? If the latter, then the dining room is "over there", so they must "get over" to it.
English had a third single-word spatial deixes, "yonder", which you can find in Shakespeare:
But, soft! what light through yonder window breaks?
It is the east, and Juliet is the sun.
-- Romeo, Romeo and Juliet, Act 2, Scene 2
And also in some regional dialects of English: https://www.theneweuropean.co.uk/having-a-ponder-over-yonder/