To give some context to the question:
Procedures of type X require manually (manual) tuning the weights...
Which is the correct form?
Manually is the adverb. Manual is (in this context) the adjective.
Tuning can be either a verb or a noun; however, in your example, tuning the weights is a gerund phrase using the verb. Here you want to modify the verb within the phrase, so use the adverb:
The procedure requires manually tuning the weights.
If instead you wanted to modify the noun tuning, use the adjective.
The procedure requires a manual tuning of the weights.
There are several ways of expressing this statement:
1.....require manually turning the weights...
which is not an idiomatic construction.
2.....require manual turning of the weights....
using the adjective to describe the gerund construction. This is an improvement but not ideal.
3.....require turning the weights manually (or) by hand...
which is the best option although still a bit awkward.
The problem is the omission of the agent who is required to turn the weights.
The statement would read most easily if you wrote:
require operators to turn the weights manually (or) by hand.
Whichever of these forms you use, the writing style is ugly and impersonal. I would write "You will need to tune the weights by hand." But of course that only works if the whole document is written in a more direct personal style.