It very much depends on the conditions of the contract that is part of the funding agreement.
Various foundations now mandate (immediate) Open Access provided by the publisher under CC 0 or CC BY license (i.e., gold open access). Other funders, such as the NIH in the US, require the publisher to deposit the postprint in PubMedCentral. As such, it really is difficult to stipulate a general rule.
If you are uncertain about your specific funding agencies' rules, I suggest contacting them. Best rule of thumb: depositing a postprint in a repository that's indexed is the minimum, but you can never go wrong with gold open access (because then you fulfill both the green and gold OA conditions).