Can you drive from Hong Kong to Mumbai?

Theoretically you can, but it is very unpractical.

First off, forget about driving directly into India from China. It is simply not possible. The Sino-Indian border is subject of a lot of military tension, and the armu are guarding the border strictly on both sides. I am very sure that civilian automobiles are not allowed unless they have a very good reason. Moreover, as the Sino-Indian border is basically the Himalayas, I am not even sure that there are a lot of paved roads in the area.

Driving through Pakistan and/or disputed Jammu and Kashmir also sounds quite infeasible, given the relationship between the two countries. You might be able to enter India via Burma (Myanmar), and quite a lot of traffic go through the Sino-Burmese border everyday.

However, there are a lot of practical problems. You need a number plate (likely) for each country you drive in [EDIT: at least for China], and also a driver's license for each country [EDIT: at least for the Chinese driver's license, you must take the written test, which is offered in Chinese only. Meaning that driving through China is not an option unless you have a reasonably good command of the language]. You will pass through a lot of areas with poor road conditions and/or unrest: in the Chinese segment, there is none AFAIK, but around the Sino-Burmese border, and in parts of India, there is the problem of (1) bad infrastructure and (2) unrest. It is not necessarily safe to travel through these areas, and even if you manage to travel safely it would be a huge pain. The entire trip might take very long due to road conditions, etc.

EDIT: I am quite sure there's no direct highway from China to India. It's hard to prove a negative, but the only references I've found about highways connecting China and South Asia are the old Stilwell Road (which actually connects China and India via Burma), and the newer Sino-Nepal "Friendship Road". There are actually three stretches of the Sino-Indian border (excluding the border between China and disputed Kashmir): the Ngari-Himchal Pradesh/Uttarkhand border (where there's least tension, but still a lot), the Sikkim border (where a border standoff happened last year), and the Arunachal Pradesh border (which is subject to a bitter border dispute). The greatest problem here is that foreigners are not even allowed into these border areas in Tibet (due to the high military pressure here), so it completely rules out the possibility of driving directly from China to India. Since, as a foreigner, you'll not be allowed into the frontier zone in Tibet anyway, you can't take the Sino-Nepal Road either. Which leaves you with the only option of going through Burma.