I know of two countries, with real-life cases: Korea and Singapore. Both in Singapore and Korea, a man can not renounce his citizenship until he has done his military service – even if he already has another passport.
In Korea it is even more perverse. A young Korean-X man, say Korean-American, can think he is American only, and still be considered a Korean citizen by the Korean government if someone in the family (usually it's the paternal grandpa) registered his birth on the family register. That makes him automatically Korean. I know of cases where KorAm dudes came to visit Korea during summer and went home 2~3 years later, extremely fit and fluent in Korean...
There is a procedure, which changes, to defer military duty until you're too old to do it. A friend of mine even managed to live in Korea, unmolested, back in the days when Korean-X kids were shipped to the Army for three years. But you have to be careful.
As for Singapore, the problem is doubly problematic: dual citizenship is illegal, and NS is compulsory. Which means you cannot go to Singapore, ever, if you're an illegal draft-dodging dual citizen.
But that's basically only applicable to a country – a Singaporean-American draft dodger can use his American passport to go to any country, except Singapore.