到(dào)期(qī) means ‘due’ and 过(guò)期(qī) means ‘expire/to exceed the time limit’.
到(dào)期(qī) is prone to be used in the field of loan, contract etc, while 过(guò)期(qī) is more likely to be used in the field of food and product. For example,
Hétong jíjiāng dàoqī
Contract be about to expire
‘The contract is about to expire.’
Shípǐn jíjiāng guòqī
Food is about to expire
‘The food is about to expire’