The B4270 and other local roads are affected, Vale of Glamorgan council has said.
The blaze at Llandow Industrial Estate started on Thursday, and South Wales Fire and Rescue Service crews are still on site.
Residents are also being advised to keep windows and doors closed.
A fire service spokesperson said: "Our crews are monitoring the fire and will do so over the coming days to ensure that the incident is contained."
Officials said no asbestos is believed to be at the site.
However, Public Health Wales (PHW) said substances in the smoke could cause coughing, wheezing, breathlessness and chest pain.
People with asthma and other respiratory conditions may be "particularly susceptible" to the smoke and should carry and use their medication.
Andrew Kibble of the PHW environmental public health team recommended people stay out of the smoke and if at home, close doors and windows.
The council's emergency planning team is working to limit disruption.
Local schools have been advised to open as usual, and there is no impact on the local water supply.
Drivers have been warned to take care as smoke from a 2,000-tonne rubbish fire near Llandow in the Vale of Glamorgan is being blown over roads.