Is it really problem if 0,3 voltage more than required applied in digital multimeter?
The batteries will be just fine in the multimeter.

Batteries are specified at a "nominal" voltage, they have a discharge curve so the terminal voltage varies throughout it's useful life. 

Here is the datasheet for a Duracell AA battery.
It's discharge characteristics show that it will be at 1.6 V when new, and the terminal voltage drops with both discharge rate and capacity used.

Devices that use these types of batteries are designed to accommodate the voltage variation. They do not necessary consume more power when the voltage is higher (new batteries) since P= V*I and the current drawn by the multimeter at any particular terminal voltage is defined by the design.