Officers say Harris Binotti took a flight to Thailand on Saturday and they have requested help from Interpol.
Police said the body of a 47-year-old man, named locally as Gary Ferguson, was found with chest and head injuries at Mr Binotti's apartment in Yangon.
The dead man's wife is said to have gone to the property when he failed to return home.
The two men are reported to have been out drinking together on Friday night.
They were both working as English teachers at the Horizon International School in Yangon, in the south of the country previously known as Burma.
A spokesman for the school said Mr Ferguson had worked there for a year. Mr Binotti, who is thought to have connections to Dumfries and Aberdeen, had been employed for the past three months.
Mr Binotti's neighbours said they heard shouting and banging noises - as if a fight was taking place - in the early hours of Saturday morning.
A spokesman for the Foreign Office said: "We are providing help and support to the family of a British national following a death in Rangoon, Burma, and are in touch with the local authorities."
Police in Myanmar are searching for a 25-year-old British teacher after the death of a colleague also from the UK.