Does a spinning disk weigh more than when it is stationary? Relativity tells us that mass increases with velocity, does this mean the faster a disk spins the more it will weigh? If so, is this measurable with current technology?
 A: 
Relativity tells us that mass increases with velocity,

This is the relativistic mass that is no longer used because of the confusion the variable "velocity" introduces. Special relativity is studied with the four vector algebra , which makes all observations covariant with Lorentz transformations from inertial frame to inertial frame . The spinning disk , when at rest has an invariant mass given by


The $m_0$ for a rotating disk will be the same in any inertial frame, and will be larger from the $m_0$ of the disk at rest, and the more energy in the spin the larger.

is this measurable with current technology

I do not think so, it will be a very ver small effect in comparison with the disk's rest mass.
A: Yes, a spinning object has more mass than the same object at rest. Any energy you pump into a system increases its mass (in its centre of mass frame), but when you're dealing with mechanical or electromagnetic energy, the amount of energy that's storable is minute compared to the original mass of the system (before you started pumping energy into it). It's only when nuclear forces are involved that the energy becomes significant.
To get relativistic effects, you generally need velocities that are a significant fraction of c, the speed of light. In Newtonian mechanics, kinetic energy is given by
$$E_K = \frac12 mv^2$$
A relativistic version of that equation is
$$E_K = \frac1q mv^2$$
When $v$ is small, $q$ is close to $2$.
This graph (from my answer here) shows the relationship between $\beta=v/c$ and $q = \beta^2/(\gamma-1)$.

Even when $\beta=\frac1{50}$ we see that $q$ is still very close to $2$.
However, it's not possible for normal materials to spin anywhere near the speed of light. Spinning bodies are limited by the speed of sound. From Wikipedia:

Any rigid object made from real materials that is rotating with a transverse velocity close to the speed of sound in the material must exceed the point of rupture due to centrifugal force, because centrifugal pressure can not exceed the shear modulus of [the] material.

The speed of sound in solids is faster than the speed of sound in air, but it's still a tiny fraction of the speed of light.
The current world record for the fastest spinning object is held by this nanoparticle made of silica (sand, quartz), which is levitated and spun using lasers to achieve a rotation rate exceeding $300$ billion RPM = $5$ GHz.


The fastest-spinning object ever created is a nano-scale rotor made from silica at Purdue University. This image of the rotor at rest was created using a scanning electron microscope. For scale, the yellow bar in the image is $200$ nanometers.

The radius of the balls composing the nano-rotor is ~$80$ nm. The speed of sound in quartz is ~$5800$ m/s, which is around $c/52000$, and the nano-rotor is getting close to that speed.
The (rest) mass of the nano-rotor is ~$5.7×10^{-15}$ g. Its kinetic energy is ~$5.0×10^{-11}$ J, which is ~$10^{13}$ times smaller than the rest energy. So to measure the mass increase, the measurements need at least $13$ digits of precision. That level of experimental precision is very difficult to achieve.
Our most precise measurements are the time measurements performed using atomic clocks. A good caesium atomic clock has precision around $10^{-13}$ s, and the latest generation of optical frequency atomic clocks can measure time durations with better than $10^{-16}$ s. So if the lasers used in the nano-rotor experiment are regulated by the latest optical atomic clocks, it might be possible to detect the effect of the rotor's spin on its mass.
A: Sure. You can't spin disk in such a way that it's tangential velocity $v_{\perp}$ would exceed speed of light, hence Lorentz factor could be re-stated for a spinning disk :
$$ \gamma = \left( 1-{\frac {\omega^2 r^2}{c^{2}} } \right) ^{-1/2}  $$
As for illustrations, fastest spinning neutron star is known to spin at $716~Hz$ rate and is of $\lt 16~km$ in radius. So this gives for quickest known pulsar Lorentz factor, $\gamma \approx 1.000\textbf7$. If we would like to "set goal" for this pulsar for increasing it's mass by $2\times $, solely by spinning faster, then it would need to rotate at amazing $16~\textbf{kHz}$ frequency! Thus, it's very hard to accumulate energy into mass, solely by object rotation.
