If I recall correctly, Yoshi's message only appears when you first get him and
you're on the Yoshi's Island submap, so a quick workaround would be to move your first map somewhere else.
To disable the rescue message more thoroughly, it looks like you'll need to dip your toes into applying patches. The ROM Map has this to say:
Originally posted by ROM Map
$01EC36 - Change from "D0" (BNE) to "80" (BRA) to disable the Yoshi rescue message.
which translates into the following patch:
Save that as an .asm file, apply it to your ROM with Asar
, and you should be good to go.