What would happen if the Earth was in a polar orbit around the sun?

If you leave the spin angular momentum unchanged with respect to the Sun's spin angular momentum then this has drastic effects. Whilst a "day" is still 24 hours, the periods of daytime and night time become much longer. I think the effect is equivalent to changing the inclination of Earth's spin to its orbit to 67 degrees. Thus anything above 23N or below 23S would be considered "polar" and would experience 24 hour illumination during some parts of the year. Equatorial latitudes would still experience a 24 hour diurnal cycle but the height that the Sun got above the horizon would vary with a much bigger amplitude.

One of the main changes would be in the Earth's geocoronal environment. The magnetic activity of the Sun is concentrated toward low solar latitudes - this means that during solar activity cycles, then every 11 years or so the Earth is subjected to a considerable bombardment in the form of accelerated charged particles and coronal mass ejections (CMEs). However the solar wind character would also change throughout the year. The polar regions of the Sun emit a thin, fast wind, whereas the equatorial regions are dominated by a slower, denser wind.

If the Earth was on a polar orbit, then major magnetic-related events, such as CMEs would be much less frequent - these are confined to latitudes $\pm 30$ degrees for most of the solar cycle, but can extend to $\pm 60$ degrees near solar maximum (Webb & Howard 2012). The "space weather" environment could be considerably quieter for several months of the year. This would be of considerable significance in my field (astrophysics) in terms of the kinds of satellites, detectors and observations one could make with lower interference from solar activity. It would also have significance for dosage of radiation that is received by astronauts working for lengthy periods of time in low-Earth orbit. We'd also miss out on a number of northern lights displays!

the Earth was in a polar orbit around the Sun?

If you took the Earth's current orbit and rotated it so that the Earth's orbit took it over the Sun's poles (with respect to the Sun's axis of rotation). The Earth's current orbit is approximately in the plane through the Sun's equator.

This is called orbital inclination. Earth's is about 7 degrees (to Sun's equator), dwarf planet Eris has an inclination of about 44 degrees (to the ecliptic)

Would the equator be perpendicular to where it is now?

That would depend on what events caused the planet to get into that orbit. Some planets don't have their axis of rotation aligned very closely to the axis of orbit.

Object  Axial Tilt in degrees
Sun     7.25    
Mercury 0.03    
Venus   178.36  
Earth   23.44   
Moon    6.68    
Mars    25.19   
Jupiter 3.13    
Saturn  26.73   
Uranus  97.77   
Neptune 28.32   
Pluto   122.53  

The tilt is shown by the arrows through the various bodies in this diagram:

- NASA

It is worth noting that the tilt (or "obliquity") isn't constant. Some planets vary a lot, Mars varies from 15 to 35 degrees and over tens of millions of years may have varied between 0 and 60 degrees. The obliquity of Mars is often referred to as being in a chaotic state.

As well as time zones?

Time zones are a reflection of the Earth's rotation and not it's orbit. They are also a political invention.

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• Planetary fact sheets
• The Chaotic Obliquity of Mars

I'm surprised no one's mentioned the effect on Earth's 24-hour days...

If this is what you mean by a polar orbit:

Except the Earth in the picture is the sun, and the satellite the Earth, and the Earth still rotates "horizontally," then the most noticeable effect to humans would be that the rotation of the Earth no longer creates 24-hour days! Half the Earth would be exposed to the sun for 6 months (creating a 6-month "day") while the other half would be shrouded in darkness for equally as long (ignoring tilt). I imagine life as we know it could not survive the temperature variation, then.

That answers your question about timezones, sort of :-)

Meanwhile the equator is by definition the plane perpendicular to the axis of rotation going through the center, so no, the equator doesn't move, relative to the Earth.