The Scottish Government said the advice also applies to other education settings that would normally be closed on a bank holiday.
The funeral is due to be held at Westminster Abbey on Monday, September 19, and is expected to be attended by leaders and dignitaries from around the world.
Major retailers, including Morrisons, Marks & Spencer, Lidl and Asda, have all confirmed they will shut on Monday.
The National Trust for Scotland said buildings and gardens will be closed while countryside places remain open on the day of the funeral.
Major galleries including the National Galleries of Scotland will also shut, as will the National Museum of Scotland.
A Scottish Government spokesman said: “The Scottish Government has advised local authorities that schools across Scotland should close on September 19 as a mark of respect.
"This advice also applies to other education settings that would normally be closed on a bank holiday.”
Some councils, such as Glasgow, had already announced closures.
Schools were already set to shut in some areas due to a public holiday weekend from Friday to Monday.
A spokesman for Transport Scotland confirmed ScotRail and Caledonian Sleeper services are expected to operate as normal.
Plans for cross-border services remain to be confirmed, “with options to increase capacity in and out of London on some routes”.
Strathclyde Partnership for Transport and Edinburgh Trams confirmed they will be running a normal service.
Ferries are also expected to continue as normal.