Well, the Devil May Cry series have been rather linear and it's worked so far. Making it a sandbox game would be a big step and I am not sure if it would work, but it would definately give us some more freedom. It might work, as long as it doesn't ruin the story. However, I prefer it to be linear.
I've noticed that many people are mentioning the Styles they used in DMC3 - personally, I'm not a big fan of it. I'd rather you have perhaps three styles available at all times, say Trickster, Gunslinger and Swordmaster, and then you could earn points which you can put in either style to enhance them - perhaps a bit like in an MMO such as World of Warcraft, where you simply specializes in a certain aspect. In DMC3 I hated that I couldn't dodge if I didn't have the Trickster style. The dodge system was better performed in DMC2, simple and easy to use.
I'd also like to see Dante evolve as he progress, giving him more powers and such instead of having him be able to do everything from the start. And have it be part of the storyline that he evolves. And the same with his mystical weapon, enhancing it with new forms - we've seen a spear form, a halberd form and a whip/grappling hook thingy - which I think is kinda cool. I've always been a fan of having a variety of weapons which do different things.
As for the guns, I'd like to see something else happening than just getting a shotgun and a bazooka... The Pandora in DMC4 was kinda cool, but also extremely weird to use. I loved the Lucifer though, because it was different.
Quicktime events... well, every other game and their mothers have it in their gameplay these days, and Ninja Theory uses it too, so I believe we might see it appear in the new DmC. It even appears in Bayonetta which is created by DMC's original creator, so yeah, unless Capcom decides to be different, I am sure there will be quicktime events.
Blood Palace must return! It was an awesome way to just smash through everything, and we should get some rewards for going through it like extra costumes and the like. In Dante's Inferno (the game) there's something alike called 'The Trials of Sct. Lucia' which is actually rather cool and it gives the players the opportunity to make some trials for themselves which they can send out online for others to download. I'd like for something like that to be in DmC as well.
And multiplayer function! The 'Trials of Sct. Lucia' is a multiplayer function, and perhaps they should make it available for Bloody Palace to be a multiplayer mode as well. I would love to see other characters returning, such as Trish, Lady, Lucia, Vergil and Nero, and have them be playable in some way. I'm not sure if a multiplayer function should be able in the storyline mode, but again, as long it doesn't ruin the storyline, then it could be cool.