I'm certain they have access to certain information of things that have yet to come to pass. After all, it is the Lord who allows these lying spirits to deceive people, and He knows the end from the beginning.
However, I do not believe they have any ability to see in the future on their own, and I believe it was the crucifixion itself that shows this in action.
You ever wonder why Satan entered Judas to betray Him to the priests, when it was the motive of Satan to prevent the crucifixion? We see this motive of his in Matthew 8:
"From that time Jesus began to show his disciples that he must go to Jerusalem and suffer many things from the elders and chief priests and scribes, and be killed, and on the third day be raised. And Peter took him aside and began to rebuke him, saying, Far be it from you, Lord! This shall never happen to you. But he turned and said to Peter, Get behind me, Satan! You are a hindrance to me. For you are not setting your mind on the things of God, but on the things of man."
Matt 16:21-23
Satans will was to hinder the crucifixion, because he knew it was at the cross he'd be defeated. So why did he enter Judas to betray Jesus on the feast day? For the same reason the priests DID NOT want to kill Jesus on the Passover. The priests feared the people would revolt against them, stopping the execution, because these were the same people who just praised Him as King just a few days before:
"Then the chief priests and the elders of the people gathered in the palace of the high priest, whose name was Caiaphas, and plotted together in order to arrest Jesus by stealth and kill him. But they said, 'Not during the feast, lest there be an uproar among the people.'
Matt 26:3-5
Satan desired to hinder the crucifixion, and planned to use the people to do it. He reasoned as the priests had, that the people would rise up to save Jesus from His execution, but they were both wrong, to the glee of the priests and to the horror of the demons. Had he been able to see the future, and know this plan would backfire to the point of him sealing his own doom, he wouldn't have done what he did on that occasion.
To go a step further, I believe it coud have been Satan who set it up so that Pilate's wife had tortured dreams (Mt 27:19), to warn Pilate agains killing Jesus, so Pilate would , he reckoned, be pressured by the people to release Jesus on one side, and pressured by his wife to do the same on the other side, because the satanic goal was still to NOT kill Jesus, a reason why he offered Jesus the kingdom of the world without the need for the cross.
It certainly wasn't God who gave her those dreams as it please God to kill His Son in our place, and Jesus stated it was for that very reason He had come.
So, if I have no reason to believe Satan can see the future on his own, I have no reason to believe his minions can either, being lesser and fallen spirits.