You are a:

Yule log

I am a body part in flames

Logs are used to build fires

The earth and I move the same

Served, typically once a year (Christmas), the earth goes around the sun once a year.

I am a dessert without tea

It is a dessert and has no 'T' in it.

When it comes to cake, I'm not easy

This is a fairly complicated cake to make that requires a complicated mix for the filling, then it's rolled, and then often decorated to look like a log.

You are a:

Yule log

I am a body part in flames

From dictionary: a large log traditionally burned in the fireplace on Christmas Eve.

The earth and I move the same

The log is burned, and the dessert is served, typically once a year (Christmas), and the earth goes around the sun once a year.

I am a dessert without tea

It is a dessert and has no 'T' in it.

When it comes to cake, I'm not easy

This is a fairly complicated cake to make that requires a special mix for the filling, then it's rolled, and then often decorated to look like a log.

I'd rather not get into the specifics, but are you?

Uranus?

You are a:

Yule log

I am a body part in flames

Logs are used to build fires

The earth and I move the same

Served, typically once a year (Christmas), the earth goes around the sun once a year.

I am a dessert without tea

It is a dessert and has no 'T' in it.

When it comes to cake, I'm not easy

This is a fairly complicated cake to make that requires a complicated mix for the filling, then it's rolled, and then often decorated to look like a log.