Sony Interactive Entertainment confirms that PS Plus Premium will not support DLC and add-ons when streaming titles.
Basically, since only PS3 games are the only ones that will be streamed, players will have to enjoy playing them without the add-ons and DLCs that should be available for purchase or to be redeemed. They will have to play these games in its vanilla form.
Please note, DLC and add-on content is not supported when streaming a title.
Actually, this is not new for subscribers of PS Now as this is quite similar to what they are experiencing before. It seems SIE has not made changes to make DLCs available and working while streaming.
This is also happening to PS4 games that are going to be streamed instead of being downloaded. It is quite the benefit that SIE gave us the option to download the game to the hard drive rather than be forced to play it while streaming, so DLCs will work. Streaming it though will have the same effect: DLCs are disabled.
Some PS3 titles only have DLCs that are cosmetics and little content packs, but there are others that have expansions. Unable to play those extra big chunks of content due to this limitation is going be one of the factors for subscribers to be on the fence on upgrading to a higher tier, unfortunately.