Can I use downloaded games from my PS3 on PS4?

No, the PS4 is not backwards compatible, meaning that you can't play PS3 games on PS4. There are talks of, maybe, the PS4 becoming backwards compatible (as stated in this article) after Microsoft announced that feature for the Xbox One. But as of now and, most likely, for some time to come you can't play games for the PS3 on PS4. However, there is an option to upgrade from the PS3 version of Minecraft to the PS4 version, for more info look HERE.

UPDATE(concerning upgrading Minecraft from the PS3 version to the PS4 version): As the link I provided now states:

As of February 25, 2016 this upgrade offer and the DLC content transfer offers are no longer available.

If you currently own the PS3 version of Minecraft, which I'm assuming you do, you can upgrade for $4.99. However, as DGarvanski stated, the PS4 is not backwards compatible at the moment. If you are interested, you can continue and pick up where you left off on your PS3 Minecraft worlds on PS4 via Minecraft's world transfer feature. This lets you send a save file to PS Vita, or PS4.