heat - How hot must a star get before it is considered to be a star?

From a physics perspective

From a physics perspective an object is a star when it is undergoing nuclear fusion, generally of hydrogen atoms at its core, this is regardless of its temperature!

A star is not determined by its temperature, it is instead determined by it's internal processes.

This does mean that if Jupiter began nuclear fusion it would be considered a star, albeit a minuscule one.

In this case it is a yes/no distinction of if an object is a star.

From an observational point of view once something is classified as a star there are 7 groups it can fall in to determined by its features.

Sourced From: http://en.wikipedia.org/wiki/Star#Classification

Class Temperature
O: 33,000 K+
B: 10,500–30,000 K
A: 7,500–10,000 K
F: 6,000–7,200 K
G: 5,500–6,000 K
K: 4,000–5,250 K
M: 2,600–3,850 K

Note: Three more classifications L T and Y have been added to the colder end of this list, but I am unsure of the cut off points so omitted them.

But strangely they are not classified by temperature but by their spectrum, it just so happens that their spectrum correlates to their temperature! The temperature spoken of here is of the photosphere of the star (where the photons begin free streaming), not its core (where photons are created from ongoing fusion reactions).

Dwarf stars have their own classification system prefixed by the letter D though.

Quote from Wiki article:

White dwarf stars have their own class that begins with the letter D. This is further sub-divided into the classes DA, DB, DC, DO, DZ, and DQ, depending on the types of prominent lines found in the spectrum. This is followed by a numerical value that indicates the temperature index.