Q:

Negative impact of having the launch pad away from the Equator

I imagine a satellite meant for reaching a geostationary orbit will benefit most from it, but being near the Equator is surely an advantage to any space launch (right?).  
However, what's the impact of each degree south or north away from the Equator on the launch? At what point would a launch become unfeasible?
Obviously, there are other factors. Only 13 countries are crossed directly by the line: Ecuador, Colombia, Brazil, Sao Tome & Principe, Gabon, Republic of the Congo, Democratic Republic of the Congo, Uganda, Kenya, Somalia, Maldives, Indonesia and Kiribati. Most of these are surely not an option. So, I get that any space port would strive to approach the Equator, but might not get too close.

A:

However, what's the impact of each degree south or north away from the Equator on the launch? At what point would a launch become unfeasible?

The rotational speed of the Earth's surface at the equator, and thus the "free" velocity you get from launching from there into an equatorial orbit, is about 463 m/s. The total velocity budget needed to get to low Earth orbit varies with the design of the rocket, but is usually around 9400 m/s, so you get about 5% of your speed for free. The rotational speed is proportional to the cosine of the latitude; Cape Canaveral at 28.5º north is thus moving at ~407 m/s, so still gaining a lot from the rotation. At no latitude is an orbital launch unfeasible; you just need a slightly more powerful rocket or a slightly smaller payload to launch from higher latitudes. Baikonur is at 46º N (321 m/s) and Plesetsk is at 63º (210 m/s)!
It does require additional energy to launch into a lower inclination orbit than the latitude you're starting from: if you want to orbit directly over the equator from Canaveral, you have to start as if you're launching into a 28.5º-inclination orbit, then modify your direction of flight as you approach the equator. 

A:

I imagine a satellite meant for reaching a geostationary orbit will benefit most from it, but being near the Equator is surely an advantage to any space launch.

Actually, no, not any space launch. 
For retrograde orbits ("backwards" orbits with inclination greater than 90 degrees) you have to overcome the spin of the Earth. Most retrograde orbits are close to polar (around 98 degrees for sun-synchronous) but sometimes they are strongly inclined, way past 90 degrees. In this case, being farther from the equator is an advantage, up until your latitude reaches the inclination itself. For example, to get to 135 degrees inclination you can launch between +/- 45 degrees without a big problem, but beyond that you have do waste a lot of delta-v to do a plane change.
You can read about an example of a strongly retrograde orbit in The strange orbit of Ofeq 11 - how does it (actually) do this?
Which leads to the second point:

However, what's the impact of each degree south or north away from the Equator on the launch? At what point would a launch become unfeasible?

It's probably never unfeasible, but latitudes larger than the inclination will become increasingly wasteful due to the required plane change. At some point you'll need a bigger rocket for a given payload mass.

