This hasn't happened to me before and I'm worried. I have purchased the expansion pass, and I have PlayStation Plus. This has happened to my ex-friend five months ago, but I cannot remember what he did to fix it. I can go into patrol, strikes and the daily missions; But I cannot go into the weekly heroic, nightfall, daily crucible or the raids.
This issue could result from several reasons, some that would already not apply to the specific case listed by OP.
The Taken King requirement increases
With the release of The Taken King, a lot of activities had their minimum requirements increased. Crucible now requires level 10, while dailies and weeklies have higher respective light requirements.
"Online Play" permission issues (Xbox and PS4 only1)
All listed 'problem activities' appear to require PlayStation Plus or Xbox Live. As a result you will not be able to access them, unless you have a valid subscription, in most cases. OP reports having PlayStation Plus, but I would recommend confirming your registration is live. You can do this by attempting to manually join a strike that attempts to place you into matchmaking, or be deliberately attempting to access the crucible.
It is also entirely likely that there is problems registering your permissions to access the service, even when you have an active subscription. Try turning your machine off and back on again. I have seen this problem arise commonly, but it appears to fix itself.
Given that all problem dailies are associated with player interaction, it is a possibility that Bungie have banned your account from accessing the service. This is not something they will do without warning, so you will likely be made aware that a 'ban' is in place. Bungie will generally do this if you act in an unfair manner. That includes using exploits or mods, taking advantage of another player's use of exploits or mods, and impersonating Bungie. You can find out more on game banning at Bungie's FAQ.
1Should not apply to PlayStation 3, which does not require PlayStation Plus for online to play
What EXP and Light levels are you?
I recently finished the "main story" of The Taken King expansion myself and a lot of stuff is locked until you hit level 40. A bunch of stuff is not even visible.
As soon as you reach EXP 40 a lot more of the content shows up. Even then you need to be a certain Light Level to actually play some of them, anything Heroic for example (light level 240+ from memory).
The DLC wont work unless it is from the same region as the game.
ie. Disk for US and DLC-s from UK PSN store won't work together. The solution is to go and buy the correct region base game, either used on disk, or from PSN and all things start working. Or the correct region DLC-s. AFAIK Sony nor Bungie will not give you your money back for the wrong region content.
Also the Season pass does not automatically download individual expansions from the perspective of the PSN Store. So if you have the season pass you need to go to PSN and "buy" the specific DLC. It will be priced as zero since you have the season pass. if you are missing one of the pre TTK expansions, then this will also activate the possibility to purchase the TTK standalone addon.
And then there are the PSN realated topics - what comes to mind is that you go and check that your PS4 console is registered as your primary active console and then redownload the DLC content from PSN.