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Please don't tell him the real name of his illness.
Literal
Him [to-には] real [genitive-の] disease-name [object-を] don't-inform please.
A polite negative request — the standard frame ~ないで下さい ('please don't do X'). Two small details are worth noting. (1) The particle on 彼 is には, the dative に plus contrastive は — marking him as the recipient of the information and implicitly contrasting with other people who may or may not be told. (2) 知らせる ('to inform, let know') is the causative-style verb related to 知る ('to know'): literally 'make [someone] know.' It governs the person being informed with に and the information with を.