Which force causes an increase in tangential speed during decrease in radius of circular motion? We know that the formula for angular momentum is has the following definition:
$$L=mvr$$
If angular momentum is conserved, we can safely assume that as the radius of the mass from the axis of rotation decreases and the mass of the object remains constant, then the speed must increase in order to conserve angular momentum, and we observe this in everyday life, from figure skaters to tetherball.
How does the speed increase without the aide of a force in these cases, or is there a force involved which causes this speed to increase?
 A: First, the more general expression for angular momentum is $\mathbf L=\mathbf r\times\mathbf p$ for a point particle with momentum $\mathbf p$ with a position vector of $\mathbf r$ relative to the point in space the angular momentum is being calculated about. This expression is important because it shows that your statement

If angular momentum is conserved, we can safely assume that as the radius of the mass from the axis of rotation decreases and the mass of the object remains constant, then the speed must increase in order to conserve angular momentum

is false. For example, if the point particle has some non-zero velocity and no forces acting on it its angular momentum will be conserved while $\mathbf r$ changes, yet it's speed is not changing.
However, although it isn't stated, you seem to be considering the scenario where we have circular motion at some radius, and then the radius is decreased to end in circular motion again at a smaller radius. Well then in that case to answer the question

How does the speed increase without the aide of a force in these cases, or is there a force involved which causes this speed to increase?

Yes, the force that causes the radius to decrease will be doing work, which causes the speed to increase.
I will note that many introductory physics problems greatly simplify scenarios like this. For example, if you have circular motion and then suddenly increase the centripetal force you will not just get circular motion at a smaller radius. This is explored somewhat in my answer here.
A: It is the radial component of velocity combined with the radial force that increases the velocity.  Let's take the simple example of a planet revolving around the sun in an elliptical orbit.  Except for the apses the planet is either moving toward or away from the sun in addition to its tangential motion.
The sun only applies a radial component of force, which for circular motion is always perpendicular to the motion so the velocity is constant. But for elliptical motion there is a radial component as well as a tangential component.  The radial component of the sun's gravitational force accelerates the radial component of velocity which causes the increase in the planet's velocity.  There is no additional force that causes the planet's speed up of slow down as the planet moves toward or away from the sun.
A: Yes, here too there is a Force only that its a different kind of force called Coriolis force
So perhaps what you didn't notice here is that as soon as you start changing radius, this whole object is no longer a Rigid Body because position of particles relative to each other is now changing.
Its as if individual particles are moving radially on a rotating frame, hence the Coriolis force comes into effect.
