"I need it to lift 2 tons this time." "I don't know if it *could* do." Why do some people use "do" like this?

As far as I am aware, this is a primarily UK usage. The difference comes in how American and British english speakers handle verb phrase ellipsis, a construction in which a verb phrase is left out because it's implied or repeated. Consider this example (taken from this blog post, which contains an excellent discussion of the issue):

A: I ate all the chocolate, even though I shouldn't have [eaten all the chocolate].

In such situations, American speakers tend to simply delete the elided phrase:

I ate all the chocolate, even though I shouldn't have.

But British speakers tend to replace the elided phrase with the pro-verb do: (A pro-verb is the verbal counterpart of a pronoun, also called a propredicate. They are both part of the general category of pro-forms, or words that take the place of a specific part of speech.)

I ate all the chocolate, even though I shouldn't have done. (BrE)

Note that American English also has a pro-verb construction, but they use do so instead of do:

I ate all the chocolate, even though I shouldn't have done so.

According to this source, all three forms are acceptable in British English, while only do so and complete deletion are okay in standard American English. (There are a few American dialects that do use pro-verb do - see this article for a discussion - but the vast majority do not.)

The phenomenon occurs very freely - with almost any elided verb-phrase that's introduced by an auxiliary. Thus, in British English you'll hear I might do, I can do, I will do, I could do, I would do, etc, while the more likely American equivalents would be I might, I can, I will, I could, I would.

Returning to your specific example, we can see that it can be expressed in three possible ways, which vary in terms of usage.

I need it to lift 2 tons this time...

(1) I don't know if it can. (common in AmE and BrE)

(2) I don't know if it can do. (impossible in AmE, common in BrE)

(3) I don't know if it can do so. (possible in both AmE and BrE)

Because "do" is a regular pro-verb in English.

"Do you think this is right?" "Yes I do".

"Do you want some melon?" "Yes I do".

When there is an auxiliary such as "will" or "can" it is optional, but can still occur.

This probably developed from "do" as an emphatic auxiliary ("I do want some"), through its use as a negative and interrogative auxiliary, which eventually became compulsory for most verbs ("Do you know?" replacing earlier "Know you?")

Edit: I realise, rereading the question, that you are asking about the doubt implied by the usage. It usually (though not always) has a certain doubt about it, but the meaning is not

I don't know if it can

but rather

It can, but there might be a better way to do it.

So it is still asserting the statement, but there is an implied "but" or "if" following.