This can be confusing because there is no authoritative body that sets terminology for Agile and, therefor, there is not "right" answer. But, these definitions and distinctions should be close.
The roadmap provides stakeholders with a view of how the product might roll out. Many teams do a planning that is longer out than a sprint but within about a quarter. The roadmap often indicates what major features are in that time period and their likely order. Though things change from sprint to sprint, this near-term period is the most likely to have minimal change. The roadmap then often describes the "next up" features. For a new bank teller application, it might look something like this:
Q(n) Focus: Account History. Likely features: Account overviews,
transaction history, transaction search, cross-account reference
Q(n+1) Focus: Account Settings (details changes and switch account
Q(n+2) Focus: New Account Creation
Visually, this can take many formats. The right amount of detail would give stakeholders and idea of where a given feature might fit over the next year and no more.
Product Release Plan
This could be synonymous with roadmap. If so, use is as above. However, I've also heard this refer to a go-to-market strategy. This might actually be less detailed than the roadmap, if you can imagine. However, it would also have key market information in it. For example, if the first release would only serve customer segment X, my Product Release Plan would call that out and include information on how we reach out to those customers without confusing others. Then, when release 2 also incorporates customer segment Y, there will be new market strategy needs to loop in. In this case, this is a document used to coordinate efforts between teams and maybe whole departments.
Minimum Viable Product (MVP)
This term has a definition from The Lean Startup and is the smallest amount of work that must be done to learn something about your customer. These are incredibly small and usually delivered in a sprint or two (tops). However, the common use of the term does not match this definition. Instead, the common use of this term is the smallest amount of product that is viable in the market. This would be that first release from the release plan I mentioned. The real term for this is Minimum Marketable Featureset, but as I said, the common use of the term is all mixed up.
As an example, if I were creating a streaming service like Netflix, a true MVP would be something that lets people simply go to a page and stream a movie - probably with a basic HTML tag. The goal here would be to see if people go to it and actually watch a movie online. (we take this for granted now, but when Netflix was created, a lot of people thought no one would watch movies online)
An MMF would involve a small library of content, some sort of browsing, user accounts, and payments. These would be nowhere near as robust as we think of Netflix now, but they would be there.