Step 1: Restate the condition to check for each person.
We need persons whose festival list contains both Christmas and Onam, and contains neither Dussehra nor Holi.
Step 2: Test each person one by one.
A: Holi, Dussehra, Diwali, Guru Nanak Birthday, has neither Christmas nor Onam, and it does have Holi and Dussehra, so A fails.
B: Shivratri, Christmas, Onam, Eid, has both Christmas and Onam, and has neither Dussehra nor Holi, so B passes.
C: Holi, Shivratri, Christmas, Diwali, has Christmas but not Onam, and it does have Holi, so C fails.
D: Holi, Dussehra, Guru Nanak Birthday, Eid, has neither Christmas nor Onam, so D fails.
E: Christmas, Diwali, Onam, Guru Nanak Birthday, has both Christmas and Onam, and has neither Dussehra nor Holi, so E passes.
Step 3: Identify the qualifying pair.
Only B and E individually satisfy the condition, so the pair that celebrates Christmas and Onam but not Dussehra and Holi is B and E.
Step 4: Rule out the other options.
A and C fail the test entirely, B and D includes D which fails, so only "B and E" is fully correct.
Final Answer:
The pair is B and E. \[ \boxed{B\ and\ E} \]