There is no indication in halacha that there is a requirement to do so. However, there are a few later authorities who cite a similar custom and debate its merits.
Before we look at this custom, it is important to note that, as a general rule, anything which is designated for use by a Sefer Torah should not be used for other purposes. However, the Shulchan Aruch and Rema (154:8) write that we use the Ark for things connected to the Torah, such as covers, silver and even other holy scrolls (megillot, etc.) because of an assumed stipulation when building the Ark. Nevertheless, one should still not use it for anything else.
The Taz (154:7), writing in the 17th Century cites the practice of putting a lit candle in the Ark on Hoshanah Rabbah when all Sifrei Torah are removed (we only to remove seven) and questions whether this is correct. He points out that in principle putting a candle in to honour the Torah should fall within the parameters outlined above and therefore one would think it is ok.
However, he notes that the stipulation for using the Ark only works while the Sifrei Torah are present in the Ark because it maintains a high level of sanctity. Once the Sifrei Torah are removed, to put something else inside lowers the sanctity of the Ark and therefore he suggests it should not be done.
A slightly later source, The Eliyah Rabbah (Siman 669), published in the early 18th Century, quotes the same practice (putting a candle in the empty Ark) and does not raise any concerns about it.
Perhaps the strongest opposition to the practice comes from the Kitzur Shulchan Aruch (138:11) who says it is a bad custom that should be stopped.
I have never seen in print the suggestion to put a chumash in the Ark – perhaps it developed as health and safety clamped down on the use of candles! It is possible to suggest that a chumash is better from a halachic standpoint as well since it may have a similar status to a pasul (invalid) Sefer Torah which we certainly house in the Ark.
In conclusion, it is not a custom we would necessarily encourage taking up but those who already have the practice seem to have authorities to rely on.