How can a magnetic field accelerate particles if it cannot do work?

A varying magnetic field generates an electric field, and an electric field can do work on a particle. This is called Faraday's law of induction:

$$\nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}$$

The full Lorentz force equation is

$$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$$

So for example, if the magnetic field is increasing in the $\hat{z}$ direction, such that

$$\vec{B} = b t \hat{z}$$

and

$$\frac{\partial \vec{B}}{\partial t} = b \hat{z}$$

then the electric field is determined by

$$\nabla \times \vec{E} = - b \hat{z}$$

Thus the electric field is not zero, so work can be done on a charged particle as a result of a changing magnetic field.

The force on a charged particle is called the Lorentz force, and it is give by:

$$ {\bf F} = q({\bf v} \times {\bf B}) $$

where the $\times$ symbols means a cross product. This means the force ${\bf F}$ is always at right angles to the direction of motion ${\bf v}$, and therefore the work done on the charged particle is zero. The Lorentz force can accelerate the particle by changing the direction of its velocity but it cannot change the magnitude of the velocity.