Ice Climbers would be absolute history
DDD's fat a** would most likely end up in Low tier territory
Falco could pretty much hold is own without chaingrabs although he will definetely lose his spot in the tier list.
Many other characters have chaingrab ability to but its rarely used.
Ice Climbers really are the only ones seriously affected other than DDD. Otherwise, they simply get really good damage off a grab than other characters. Get a grab at 100%-130% is a guaranteed KO on characters. Not to mention they could get grab, pummel, and nana damage. They and DDD I see dropping into the messed up mash that is mid-tier. But, I doubt low tier.
Falco's seem to hardly ever rely upon the chain grab any more these days. People know to avoid it. They do it regularly, and even then, it should be at best a damage racking until only 60% (finisher included). The SDI to avoid death has been known and practiced for awhile, so its no longer such a big deal. Its his awesome neutrals and keep away game that really keeps him up there. The loss of a chaingrab I doubt would do much to harm him. Additionally, with the lack of a chaingrab from pikachu and ice climbers, his only two bad match-ups are completely removed and they go back to being even match-ups. So, he sort of gains and losses something, and I think would neutralize to him still being one of the best in the entire game.