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.

Waitrose said like-for-like sales rose 2.8% in the five weeks to 3 January, compared with last year.
House of Fraser, which has 59 shops across the UK and Ireland, said like-for-like sales jumped 8% for the six weeks to 3 January, versus last year.
Sales in the week before Christmas were the 165-year old firm's best ever.
Both retailers' saw a sharp increase in online sales compared to last year, with Waitrose reporting a 26.3% rise in online grocery sales, and House of Fraser seeing a 31.2% jump.
Waitrose, which has 334 shops, is owned by John Lewis, which reported strong festive trading figures on Monday.
"As a business owned by the people who work here, we can take the long-term view and our Christmas results show the effectiveness of our strategy of investing in good value, in making our shops attractive destinations and in building our online business," said managing director Mark Price.
House of Fraser's chief executive John King said he was "delighted" with the chain's Christmas trading period.
The firm said it planned to continue to invest in its online offering this year, as well as refurbishing more stores.
"Given this further investment, we believe that we are well placed to continue to grow in 2015," added Mr King.
Last year, the privately held firm sold a majority stake in its business to Chinese conglomerate Sanpower, giving it an 89% share in the company.
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.
[NEXT_CONCEPT]
Upmarket supermarket chain Waitrose and department store chain House of Fraser have both reported strong sales over the key Christmas trading period.