What's the term for people who deny others' achievements? I know a bunch of people that are always in deep denial (at least trying to appear so) of other people achievements. It can be another country or another group or another religion.

For example, when you tell them, "Did you hear about Mike? He just bought a Lamborghini," they have several type of answers ready for this situation, like: "Who cares? Lamborghini was never good, you need to go to gas station every 15 minutes," or like, "But don't you know how terrible his wife is?" or, "He might have stolen someone's money," etc.

They know they are wrong, but they say these types of things to ease their deep internal pain. You would call them some sort of jealous people, but it's not only jealousy. Sometimes when you tell them things like, "Did you see the recent car Germany just built? It just runs on salt water!" They come back with sentences like, "Whatever, I would die rather than living for a second in Germany." Everyone would be like, "WHAT?" But yes, they say these sort of things.

So what do you call these people? What's their problem? Do you have any advice to offer these people?
Regarding "What's their problem?" Question,

Long answer:
Social Psychology recognizes the concept of construals, which are how individuals perceive, comprehend, and interpret the world around them, particularly the behavior or action of others towards themselves.

Two of the basic source/motive of construal in human beings is the need to feel good about ourselves and the need to be accurate. These two needs, however, doesn't always coincide one-another; and when they don't, it builds-up anxiety inside oneself. Leon Festinger further explained those situations as Cognitive Dissonance, an uncomfortable feeling that arose from having two conflicting beliefs/ideas/values at the same time.

As stated in the linked wikipedia article, to reduce dissonance, someone may do one of these 4 things:


  
  Change behavior/cognition
  Justify behavior/cognition by changing the conflicting cognition
  Justify behavior/cognition by adding new cognition
  Ignore/Deny any information that conflicts with existing beliefs
  


In your case, your information most likely triggered a cognitive dissonance for him.

Caused by either an upward social-comparison process, which has been proved to be able to happen both subliminally and automatically in a study; and may resulted in negative-affect, especially for someone with low self-esteem and negative mood. i.e "That makes Mike better than me" or "That shows how great Germans are, but I'm not German!" An existing contradicting beliefs. i.e "I hate Mike/Germany" or "I'm far richer than Mike/My country is better than Germany." Having done or still doing something which conflicted with the new information. i.e he just bragged about his brand new car, just explained how Germany is not good, just turned down an offer to live in Germany, etc. Or a combination of all three.

Then, to reduce the dissonance, they just changed their cognition to justify themselves (Lamborghini is not that good anyway), create a new cognition (but his wife is terrible), or simply deny (he must have stolen someone's money) or ignore (whatever, not going to live in German anyway) the information. All those strategies would be easier to employ than actually changing their own behavior (buy a better car than Lamborghini, make an even better invention) or cognition (motivate themselves to earn more money or study more)

Short Answer:
You indirectly arouse his anxiety, either by lowering their self-esteem by presenting a concrete and unavoidable better social comparison target; or by presenting an information which conflicts with his beliefs/ideas/values. In order to reduce that anxiety, and so making them feels good about themselves again, people will either change their behavior/cognition, make justifications, or simply ignore/deny the conflicting information/stimuli.

Really Short Answer:
They are most likely trying to reduces the anxiety which was caused by your information