The claim "Rohingya is a Bengali dialect" cannot be meaningfully proven due to difficulties in defining what constitutes a dialect and what constitutes a language. But it is safe to say "Rohingya is a member of a dialect continuum that includes Bengali".
Rohingya is classified as a part of Bengali-Assamese languages. As such, it is not related to Burmese, which is a Sino-Tibetan language.
As said in other answers, Wikipedia also mentions that it is related to the Chittagonian language, which, according to the same source, "is often considered to be a non-standard dialect of Bengali, although it is not mutually intelligible with it".
I couldn't find any reference on the mutual intelligibility between Rohingya and Bengali, (or between Rohingya and Chittagonian for that matter) but it seems safe to assume that Rohingya is not mutually intelligible with Bengali.
The concept of dialect is hard to define. But mutual intelligibility usually plays a role. But even if my assumption about Rohingya not being mutually intelligible with Bengali is true, Wikipedia mentions that Bengali and closely related languages form a dialect continuum. That is, they are usually mutually intelligible with the neighboring languages but not necessarily so with Bengali or each other.
Therefore we can at least say that Rohingya is a member of a dialect continuum that includes Bengali.