Visiting Rio de Janeiro, Brazil as a volunteer for the 2016 Olympics what funds do I need for a visa, and in what form?

You will need to apply and obtain a Visa for Volunteer Work (VITEM I) to visit Brazil to work as a volunteer. You are not allowed to do so on a tourist visa. To apply for a visa you will need to provide the following documents:

Voluntary work (VITEM I)

1. Applicant’s valid passport – ID page;
2. Visa Request Form Receipt – photo and signature;
3. Police check certificate;
4. Proof of professional experience or qualification compatible with the work to be carried out. Any document submitted to prove this qualification, has to have been notarised (attested) by a notary public of the FCO, prior to submission to the Visa Section. Please bring this original(s) notarised, plus an ordinary photocopy of it. If the document is issued by another country, it has to have been legalised by the Brazilian Diplomatic Mission in that country;
5. Letter of invitation from the Brazilian institution (signature certified by a notary in Brazil);
6. Document with details on work place and type of activities to be carried out by the applicant (signature certified by a notary in Brazil);
7. An undertaking in writing from the Brazilian institution assuming full responsibility for the maintenance of the applicant, as well as for medical and hospital expenses of the applicant and his dependants while in Brazil and for his/her return to the country of origin (signature certified by a notary in Brazil).

Now, since this is a effectively a temporary residence visa, the regulations don't seem to specify that you need to show proof of sufficient funds for the entire length of stay. My assumption is that the authorities expect most of your expenses to be taken care of by the inviting institution. Now, I would provide a paper documenting this upon application.

In addition, I would err on the safe side of things by providing the proof of funds usually requested for tourist visas. These are:

4. Proof of financial means of support during the applicant’s stay in Brazil. Bank or credit card statements from the last 3 months at least, or latest pay slips.

Also, include the booking confirmation and invoice as both a proof of accommodation and an extra proof that you can actually pay for your stay there.