Whether he will is answerable only by waiting for the next movie.
Whether he can is another question (and the one I'll attempt to answer).
The events of the J.J. Abrams movie created a new timeline, so many of the events we've seen in the original timeline will no longer occur.
But we've also seen plenty of events that were set in motion by things that happened before the timelines split, and that would not be affected by the events of the movie, either the destruction of the USS Kelvin or the later events.
Here are some examples of threats that should still be approaching Federation space, and that Spock could warn about: The Doomsday Machine .
The Space Amoeba .
Nomad V'Ger
The Borg (though the initial encounter, forced by Q, might be affected by the changed timeline)
The Whale Probe The "butterfly effect" might prevent some of these, but it hardly seems realistic that it would prevent all of them.
(Then again, realism isn't necessarily the deciding factor.)
The Temporal Prime Directive , as mentioned in Keen's answer, might theoretically prevent Spock from revealing this information.
On the other hand, I personally hope that he'd adopt the same attitude as Doc Brown in Back to the Future :
Marty McFly: What about all that talk about screwing up future events?
The space-time continuum?
Dr. Emmett Brown:
Well, I figured, what the hell? UPDATE :
In the sequel movie, "Star Trek Into Darkness" , young Spock did consult with "Spock Prime" regarding the situation they found themselves in (avoiding spoilers):
As you know, I have made a vow never to give you information thatcould potentially alter your destiny.
Your path is yours to walk, andyours alone.
That being said, ...
followed by advice based on his knowledge of the original timeline.
He didn't literally say "Well, I figured, what the hell?", but it seems to be the same general idea.
Since Leonard Nimoy, sadly, is no longer with us, I presume old Spock won't appear in the next movie.
I have no idea how they're going to handle that.
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In the IDW comic #09 " The Truth About Tribbles " (considered a canon source of information for the new Star Trek universe according to Robert Orci)
this issue is specifically addressed by the crew of the Enterprise.
Old Spock encounters a tribble at the start of the two-parter but fails to inform the crew about the threat they pose.
Later, nuSpock advises that he's discussed whether Spock-Classic is willing to give the Federation of info about impending threats.
The answer is a definitive "no".
We can reasonably assume that this also applies to the threats you've mentioned above.
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He can't do so.
The Temporal Prime Directive prevents him from interfering in events in the past, as he's technically a time traveler.
I guess he showed me, given his warning the Enterprise about Khan in Star Trek: Into Darkness .
However, given Leonard Nimoy's death, it's exceedingly unlikely for him to appear again.
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No, because it's a new timeline (destruction of Vulcan, etc) so those events will not necessarily occur.
They may , but they will probably not.
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Keeping in mind the apparent time logic, the events that occured in the early movies will not occur again.
First, like Spock said in the movie, because the destruction of Vulcan and the explosion of the USS Kelvin created a new course in time.
This change would prevent the director from repeating the story that was told before and give freedom to be creative.
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You have to remember that the events leading to V'ger , the Whale probe and the Borg Threat start out way before Nero changed the timeline and will still happen.
V'ger is still on
it's way
and so are the Whale Probe and the Borg (since there was a signal sent from the 22nd century to the Delta quadrant).
No matter what Nero did by coming back at the time he did ( 2233 ), it won't affect those events.
Even if a new timeline was created, it's mostly a change to events immediately around Nero's arrival; it doesn't effect events happening before that.
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Spock Prime logically might believe it is a moral obligation to prevent disasters that can be avoided, such as the future destruction of Romulus, the Klingon moon Praxis exploding, the whale probe, etc.
Billions of lives could be at stake, and the needs of the many...
The fact is that the timeline is so different now, what could it hurt to try and save that many lives?
Entire planets are at stake.
Isn't that what Kirk, Spock and the rest have always done?
Why would it be any different now?
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Did Spock Classic (in the new timeline universe) warn NuSpock of upcoming natural disasters? i.e. V'ger, the whale probe, Praxis and explosion, the Borg, violent first contacts, the wave from Generations, etc?
Did Spock warn of disasters in the new Star Trek universe?
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<star-trek><history><timeline><spock><star-trek-reboot>
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 Keeping in mind the apparent time logic, the events that occured in the early movies will not occur again. Even if a new timeline was created, it's mostly a change to events immediately around Nero's arrival - it doesn't effect events happening before that. Spock didn’t warn of disasters because the new universe deals with a new timeline.
Spock didn’t warn of disasters because the new universe deals with a new timeline.