Back in June, Bungie offered some details on what Xbox 360 and PlayStation 3 owners would be able to do on Destiny after August. Apparently, "Legacy" console owners will have until 2:00 AM PDT on August 16th, 2016 to get their sweet, sweet time in on Bungie.net.
After this date, 360 and PS3 Destiny players will not be able to access their Character data on Bungie.net. So, it's in your best interest to complete the Year Two Moments of Triumph event as soon as possible, so you can check out your Record Book and get your T-Shirt code from Bungie.net before you can no longer access the page.
If you are playing Destiny on the 360 or PS3, you can keep playing the Year Two Moments of Triumph event until September 20th, 2016, but to get your shirt code you'll have to import your account to either the Xbox One or PS4 before October 4th.
There are other changes coming on August 16th though. After this date, 360 and PS3 players will not be able to purchase Silver, but will still be able to spend whatever Silver was purchased prior to that date.
The Iron Banner and Trials of Osiris will still be available through the end of August 2016, however, after that the events (as well as a number of others) will be gone for good.
Essentially, you'll need to upgrade your console in order to keep playing Destiny the way you have been since release.