持DI有急事想去外國如何辦理手續

In this case, there is 2 issues.

1. Unable to leave Hong Kong.

This portion can't be answered due to lack of information. At the least, what is the reason why your relative can't leave Hong Kong.

2. Visa

With limited exceptions, a DI holder must obtain a visa before going to another country. Each country uses different procedures. Without any information, it is difficult to answer what is needed to be done.

However, generally speaking, your relative must contact the country's Consulate and speak to a staff to make necessary arrangement for an emergency visa.

At the same time, your relative must show proof why an emergency visa is needed (for example, if it is due to life-or-death of a family member, the doctor of that family member must provide a doctor note to show the urgency).

2015-07-30 17:43:28 補充：
There is no such thing as temporary passport in Hong Kong.

Beside - even one was available, many countries place heavy restriction on temporary passports. The holder will need to get the visa anyway.

And - your relative is not qualified for a passport anyway.

非香港永久居民想出國，需持有效的DI 到該國駐港領事館申請簽證就可以了；