There's probably a neater way to do it but you could just serialize the date into a string like so:

Datetime oneyearago = System.today().addDays(-365);
String jsonDatetime = JSON.serialize(oneyearago);

Which outputs the specified date in the correct format:

2015-01-06T00:00:00.000Z

In response to your comment:

String OppJSON = '{"attributes":{"type":"Opportunity","url":"/services/data/v25.0/sobjects/Opport‌​unity/500E0000002nH2fIAE"},"Id":"500E0000002nH2fIAE","LastModifiedDate":"' + jsonDatetime + '"}';
Opportunity opp = (Opportunity) JSON.deserialize(OppJSON, Opportunity.class );

Basically just replace the hardcoded value with the variable.

P.S. There may a formatting issue there, but you get the idea.

There's probably a neater way to do it but you could just serialize the date into a string like so:

Datetime oneyearago = System.today().addDays(-365);
String jsonDatetime = JSON.serialize(oneyearago);

Which outputs the specified date in the correct format:

2015-01-06T00:00:00.000Z

In response to your comment:

String OppJSON = '{"attributes":{"type":"Opportunity","url":"/services/data/v25.0/sobjects/Opport‌​unity/500E0000002nH2fIAE"},"Id":"500E0000002nH2fIAE","LastModifiedDate":' + jsonDatetime + '}'; 

Opportunity opp = (Opportunity) JSON.deserialize(OppJSON, Opportunity.class );

Updated with a string that works correctly, but really you should look into crop1645's approach as that is more aligned with unit testing best practices.

There's probably a neater way to do it but you could just serialize the date into a string like so:

Datetime oneyearago = System.today().addDays(-365);
String jsonDatetime = JSON.serialize(oneyearago);

Which outputs the specified date in the correct format:

2015-01-06T00:00:00.000Z

There's probably a neater way to do it but you could just serialize the date into a string like so:

Datetime oneyearago = System.today().addDays(-365);
String jsonDatetime = JSON.serialize(oneyearago);

Which outputs the specified date in the correct format:

2015-01-06T00:00:00.000Z

In response to your comment:

String OppJSON = '{"attributes":{"type":"Opportunity","url":"/services/data/v25.0/sobjects/Opport‌​unity/500E0000002nH2fIAE"},"Id":"500E0000002nH2fIAE","LastModifiedDate":"' + jsonDatetime + '"}';
Opportunity opp = (Opportunity) JSON.deserialize(OppJSON, Opportunity.class );

Basically just replace the hardcoded value with the variable.

P.S. There may a formatting issue there, but you get the idea.