Does momentum conservation imply energy conservation?
While there are some good answers, no one has yet mentioned Noether's theorem. Fundamentally, the conservation of momentum and conservation of energy have different characters. Notably conservation of linear momentum is linked with the laws of physics being independent of where something happens and conservation of energy is linked with the laws of physics being independent of when something happens.
Simply: Linear momentum conservation is linked to position invariance. Energy conservation is linked to time invariance.
More rigorously, Noether's theorem demonstrates that conservation of linear momentum arises from a Lagrangian that has no explicit dependence on position and conservation of energy from a Lagrangian that has no explicit dependence on time. Furthermore, other symmetries imply other conservation laws. A lack of explicit dependence on the angle in the Lagrangian gives conservation of angular momentum, suggesting that angular momentum can be considered fundamentally different than linear momentum (which is backed up by spin in quantum mechanics not being interpretable as linear motion). As an aside, purely (Newtonian) 2-body gravitational systems have another conserved vector owing to symmetry in the Lagrangian of the inverse square force law.
Hence, it is easy to write down models where energy is conserved, but momentum is not, and systems where momentum is conserved and energy is not. Furthermore, these models can be good approximations of many real situations as described in other answers. You can have models of collisions where momentum is conserved but energy is "lost" to internal degrees of freedom or radiation. You can have thermodynamic models where you have a container of effectively infinite mass (which leads to momentum not being conserved when gas molecules collide with the walls) but with conserved energy. In the end, however, linear momentum, angular momentum, and energy seem to be conserved in fundamental interactions and maybe in the universe as a whole.
As an aside, Special Relativity treats position and time in a similar way and likewise wraps up momentum and energy in the 4-momentum, which shows energy as the time component of a relativistic 4-momentum. However, although relativity shows some symmetry between position and time and consequently momentum and energy, they still have different characters owing to different signs in the signature ($\mathrm{d}s^2 = -c^2\,\mathrm{d}t^2 + \mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2$).
The total energy (internal + kinetic) of hot body isotropically emitting radiation while moving in the vacuum far from other bodies, will decrease, while momentum is conserved.
Yeah, it's interesting to contemplate the relation between energy and momentum.
Energy can morph from form to form, with kinetic energy being only one of them. Momentum, on the other hand, deals with a subset of all forms of energy: motion.
So let's set up a system with the energy oscillating between potential energy and kinetic energy: let's say a vibrating string (for example a guitar string).
The momentum of the string is oscillating, while the total energy is conserved.
However, looking at it that way is cheating. The string is part of a system: the system as a whole is the guitar with the string. To aid this thought demonstration, take a guitar with one string, and make that string really heavy. Have that system be suspended, such that it cannot exchange momentum with surroundings. Then the string and the body are continuously exchanging momentum with each other, and the momentum of that system is conserved.
In your question, you submitted the case of an object connected to a central axis that is infinitely rigid, with the object moving in a circular motion.
That example doesn't actually offer what you want.
You need to consider the system as a whole. This means that you also need to consider the momentum of the support that maintains the circular motion of the circumnavigating object.
Consider for example the motion of the planet Jupiter around the Sun. Jupiter and the Sun are both orbiting the common center of mass of the Sun/Jupiter system. That common center of mass lies somewhat outside the Sun, not even inside the surface of the Sun.
About the title of your question:
"Does momentum conservation imply energy conservation?"
In my opinion: from an axiomatic point of view: no.
I think that in terms of formal logic the two are independent axioms. In other words, I think that in terms of formal logic neither can be derived from the other.
However, if both energy conservation and momentum conservation are laws that hold good universally, and presumably they do, then in any thought demonstration we come up with we will always see that energy and momentum are both conserved.
I think: in terms of formal logic the fact that both hold good universally does not logically imply that they are interdependent (but thought-provoking for sure).