Firstly: in all of these statements, the information that the bank will not lend us money is conveyed. The different usage of 'any more' simply implies things about the context of the statement, depending on how it is used.
'The bank is not willing to lend us any more money.'[1]
In this context, the usage of 'any more' to modify 'money' implies that the bank has lent us money at some point in the past. This usage implies this longer form: 'The bank has lent us money in the past, but [the bank] is not willing to lend us any more [money].'
'The bank is not willing to lend us any money.'[2]
This statement does not have exactly the same meaning as [1]. It lacks the implication that the bank had lent us money in the past.
The bank is not willing to lend us money any more.[3]
This statement is almost semantically the same as [1]. It is different, though, because 'any more' modifies the verb 'lend', rather than the noun 'money'. The statement has two interpretations:
- The bank has lent us money in the past, but is no longer willing to do so.
- The bank was willing to lend us money in the past (they may or may not have actually done so), but they are no longer willing to.
In the first interpretation, it is the same as [1]. However, clearly the second is not the same as [1]. It is impossible to know which one of these interpretations is the correct one without further context.
Overall, how you use 'any more' in this case depends on a few things:
- what the context of the statement is
- personal preference
- what sounds more natural (difficult if you're not a native speaker)
Both [1] and [3] are correct English, and similar enough in meaning to be used interchangeably. However, as a native speaker, [1] sounds more natural (simply because it is idiomatic - there is no rule that decides this).
Do be careful, though! 'any more' can change meanings significantly when misplaced. Compare:
- 'Are any more of your friends coming?' [a]
- 'Are your friends coming any more?' [b]
[a] implies that some of your friends are already with you, and you are being asked if additional friends will arrive.
[b] implies that you planned to bring your friends with you, but they have not come. You are being asked if any of them will arrive.
This is a big difference!