Would armor made out of molybdenum(Mo) protect from any kind of damage and would it cause any harm to the person wearing it?
No Molybdenum armour would not be worth wearing.
While Molybdenum is often used in alloys of iron, only rarely is it used on its own.
With a Mohs hardness of 5.5 Molybdenum is a bit harder than steel (4.5) it would probably have a similar protective value to steel. It would be significantly heavier having a density of 10.28 g/cm³ compared to iron's 7.874 g/cm³.
While Molybdenum has lots of interesting chemical properties all you really want out of armor is the ability to withstand a beating without deformation or shattering.
Since it is much more expensive than steel and provides similar protection, stick to your preferred steel alloy for all your defensive needs. Who knows it may have a small amount of Mo in it already.
As @sphennings explained, molybdenum is around 20% heavier, but it is also more than 20% harder.
This makes molybdenum armor more useful than iron armor.
The problems with it:
- Its melting point is 2600°C, compared to iron's 1500°C. This means, it is probably not processable with medieval technology, and it is totaly impossible for them to produce it from ores. We produce it by burning its sulfide in air, and then reducing the resulting oxide with ammonia. This has been possible with the chemical technology of the XIX century.
- If your civilization has this technology level, they can use molybdenum more effectively than iron (although it will be much more costly for them).