The common saying that one should not sit on a table because a table is like a mizbeach is partially true. The Ramo (O.C. 167:5) states that one should bring salt to the table before eating bread. Every korban on the mizbeach required salt; similarly, ever since the churban Beis Hamikdosh, every meal that includes bread at a table requires salt.
A table at which one dines is compared to the mizbeach and the bread that is eaten is compared to a korban. The analogy is based on the Gemoro (Berachos 55a), where it says that in the time of the Beis Hamikdosh, the mizbeach aided in atonement; today bread served on a table can assist in atonement. Whether by sharing a meal with those less fortunate or by learning Torah at the table, one can gain atonement similar to that provided by a korban on the mizbeach.
Thus, we see that a table is like a mizbeach. That part of the common saying is true. The prohibition of sitting on a table, however, is not found in any primary source that we have.
The Sefer Chasidim (920) tells the story of a man who used his table regularly for seforim. When he needed the table for a meal, he would remove the seforim. Once his son stepped on the table, and when he got off, a knife lacerated him. The father attributed the punishment to his lack of respect for a table that was used for sifrei kodesh. Most probably, the well-known “prohibition” you refer to is rooted in this anecdote. Technically, however, it should be limited to stepping on a table regularly used for sifrei kodesh. Besides, it is debatable whether the dangers and prohibitions mentioned in Sefer Chasidim were meant for the general public.
Although there is no source to prohibit sitting on a dining room table that is not regularly used for seforim, the mere fact that people say one should not sit on a dining table may be enough of a reason to avoid doing it. The Yerushalmi teaches that one should be careful in situations that other people consider dangerous. A person should avoid doing something that people believe could lead to a catastrophe, so the fact that people say something can sometimes be a reason to go along with it. In this case, however, I don’t think sitting on tables is generally considered dangerous behavior. It is viewed as disrespectful to the mizbeach and for that concept there is no primary source.
Therefore, there is no prohibition.
However, every parent has the right and responsibility to raise their children with the derech Eretz that they feel is appropriate. It is advised that children should be taught to behave in a dignified manner which includes no sitting or climbing on tables.
