What's the purpose of a ferrite bead inductor on this circuit?

Yes, it's basically part of a low pass L-C (or R-C for high frequencies) filter for the analog Vdd on the chip (with C9). Ferrite beads act like very low resistance (< 1 Ohm) for DC, inductors (several uH) for low RF frequencies and they act like resistors (hundreds of ohms to 1K or more) for frequencies in the 100MHz range.

Ferrite beads are normally specified in ohms in the resistive region (relatively high frequency- usually 100MHz) where they are very lossy, but they have an inductive region at lower frequencies. Note that care must be taken with this kind of circuit that noise inherent in the supply does not resonate at high Q with the ferrite and capacitor or the bead may actually result in more noise on the power rail. A good reference on beads can be found here. This sort of thing can cause all kinds of grief if the SMPS noise is (or has a harmonic) close to resonance and it changes with loading or input voltage or temperature to move in and out of resonance.

If you simply omit it the chip will not work properly since there will be no power supplied to Avcc. If you replace it with a short, the chip will operate normally, however you may see somewhat higher noise on ADC readings and possibly other subtle effects on analog performance.

R33 may have been a thought to allow for a ferrite bead in the analog ground (usually not a good idea) or it may be used as a net tie to enforce a single point connection between the analog ground nets and the ground.

Ferrite beads act as high-frequency noise filter. The beads are manufactured such that certain ranges of frequencies (typically very high frequencies) will excite current flow within the bead and ultimately dissipate as heat, effectively stealing energy away from the noise at those frequencies. Ferrite cores are what you see on most of the cables connected to your computer and TV. They are hollow cylinders that wrap around the outside of the cable to suppress noise.

The ferrite bead on your circuit is contained within a chip and put in series with the line, acting more like an inductor. But manufacturers do not typically give an equivalent inductance value in Henries. Instead, they provide a chart that shows the equivalent impedance of the bead at various frequencies.

For example, ferrite beads from the BLM-21 series in your circuit have an impedance vs frequency graph that looks like this:

You might choose one of these beads if you expect to experience EMI between 20MHz and 1GHz.

The author of that circuit clearly wanted to minimize noise on the analog circuitry of the MCU. Removing the ferrite bead (and shorting AVCC to VCC) may or may not cause inaccuracies in the analog peripherals, depending on whether noise in those frequencies are prevalent.