Are linear algebraic groups always finite-dimensional?

For the sake of being explicit, here's a simple example of an infinite-dimensional affine group scheme, even a reduced one. You might call it infinite-dimensional affine space $\mathbb{A}^{\infty}$. Its functor of points takes a commutative $k$-algebra $R$ ($k$ the underlying field) and assigns

$$R \mapsto R^{\infty} = \prod_{i=1}^{\infty} R$$

with group operation given by pointwise addition. As a Hopf algebra,

$$\mathbb{A}^{\infty} = \text{Spec } k[x_1, x_2, \dots ]$$

with comultiplication given by extending

$$\Delta x_i = x_i \otimes 1 + 1 \otimes x_i$$

(which should look familiar if you've ever written down the Hopf algebra of functions on affine space $\mathbb{A}^n$). You can think of $\mathbb{A}^{\infty}$ as a cofiltered limit of the finite-dimensional affine spaces, and this picture can be generalized to arbitrary affine group schemes.

Matt Samuel's comment did it for me: I'd misunderstood the definition of "affine variety". Originally I thought this was just a reduced scheme over $\mathbb C$ except that we only consider the closed points (i.e. look at $\operatorname{MSpec} R$ for some reduced $\mathbb C$-algebra $R$). Now I see that the various true definitions all require some finiteness condition on the variety (e.g. from a scheme point of view, we want it to have finite type over $\mathbb C$). Thanks all.