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Given the equation $$F_{\text{net}} = ma$$

Does this not imply that if the net force on a certain object is positive, its acceleration will also be positive, and theoretically this object would accelerate forever to an infinite velocity?

i.e envision a block on a surface. If a person were to apply a force (push) on this block that exceeds friction, ad continuously applies this force, this block should technically accelerate infinitely?

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Yes it does, but the law breaks down for large velocities approaching the speed of light. –  David H Dec 1 '13 at 8:46
thanks for your question, BTW you can typeset math nicely by enclosing your equation in dollar signs. –  innisfree Dec 1 '13 at 10:27
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Yes. Your analysis is completely true on the basis of Newtonian Mechanics. But upon observation of the universe this turns out to be wrong, as once the objects in consideration approach the speed of light, we have to apply Relativistic Mechanics. So, as far as Newtonian Mechanics is concerned, objects can have infinite velocity and momentum. But they can't in the real world.

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Clarification: while velocities are limited to less than the speed of light, there is in principle no upper bound on how large momenta can be. –  David H Dec 1 '13 at 8:55
$F_\mathrm{net} = p$ still holds. –  lionelbrits Dec 1 '13 at 11:44
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