In reality the vacuum energy is zero. It must be so to be in line with the measured flat geometry of space-time of the universe. Dark Energy can be explained by the cosmological constant, which is separate from the energy content of space-time.
So how then do we deal with the mathematical prediction of an infinite value? As mentioned in other answers, one such "solution" is renormalization. I put that in quotes because that process doesn't actually resolve infinites, it just moves them elsewhere. While I know most physicists wouldn't claim to be mathematicians, I can tell you that from a mathematical perspective this isn't really a valid solution. It's only hotly defended because the end result "works". But it's pretty much the equivalent of hand waving and ignoring the issue... which again is defended because ignoring it gives predictions that work.
What's amazing to me, though, is how easy it is to remove the infinity via a simple change in the program of quantization... indicating that it's existence is just a mistake of program / mathematics and not something "real".
As an example, let's look at the simplest of cases: the harmonic oscillator. The typical prescription is to write our energy equation, then convert everything to operators:
$$
E = \frac{p^2}{2m} + \frac{1}{2}m \omega^2 x^2 \\
E \rightarrow \hat{H} \\
p \rightarrow \hat{p} \\
x \rightarrow \hat{x}
$$
We then create creation and destruction operators using the position and momentum operators per the usual story, all culminating in the usual problematic equation:
$$
\hat{H} = \hbar \omega (a^{\dagger} a + \frac{1}{2})
$$
But now let's do something different. Let's change variables BEFORE the replacement with operators:
$$
q \equiv x + \frac{i}{m \omega}p \\
E = \frac{1}{2} m \omega^2 q^* q
$$
And now we replace our new variables with operators:
$$
E \rightarrow \hat{H} \\
q \rightarrow \hat{q} = \hat{x} + \frac{i}{m \omega} \hat{p} \\
q^* \rightarrow \hat{q}^{\dagger} = \hat{x} - \frac{i}{m \omega} \hat{p}
$$
If you do this, and then create creation and destruction operators again, you instead get:
$$
\hat{H} = \hbar \omega a^{\dagger} a
$$
It's worth asking here, what have we really done? Isn't it just a mathematical trick to change variable definitions before second quantization? Does this mean quantum theory "prefers" complex variables for quantization over real ones? BTW, if you apply this same trick to photons and the electromagnetic field, the infinities disappear there as well. Perhaps this mechanism is just a "better" way to renormalize the theory... where we change quantized variables as opposed to moving infinities into bare masses and charges.
From my point of view, since it's so easy to remove these infinites via a simple variable change, they aren't real / a real prediction of the theory and can, for all intents and purposes, be ignored... The only time this really matters is when trying to introduce gravity into the theory anyway.
