If I have 3 children who live with me and my Fiance and will be claimed on his return do I then still list them on my tax return but choose other parent will claim child?

No. Do not enter them at all. If you and the other parent live together, only one of you can claim the child for any tax benefit. The interview is confusing (it's designed for divorced parents). The second parent should not enter the child, at all. 

If you and the other parent live together, either one of you (but not both) may claim the child. You may decide between you which one will claim the children. You may claim one and him two, etc. Only if you can’t agree, do the IRS tie breaker rules apply, to see who has first choice. It may be worthwhile to prepare trial returns, both ways,  to see which way the family comes out best. This tool may be useful: https://turbotax.intuit.com/tax-tools/calculators/taxcaster/?s=1.

BUT, you say YOU have three children, not WE have. If the children are not related to him; then there is the question of whether he can claim them at all. He may be able to claim them as dependents for their $4050 exemption (dependent income deduction); but he cannot claim them for the earned income credit, child tax credit or Head of Household filing status.

There are two types of dependents, "Qualifying Children"(QC) and standard ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, a relationship test and a residence test. Only a QC qualifies a taxpayer for the Earned Income Credit and the Child Tax Credit.

 A person can still be a Qualifying relative dependent, if not a Qualifying Child, if he meets the 6 tests for claiming a dependent:

1. Closely Related OR live with the taxpayer ALL year

2. His/her gross taxable income for the year must be less than $4,050 (2016-17)

3. The taxpayer must have provided more than 1/2 his support

In either case:

4. He must be a US citizen or resident of the US, Canada or Mexico

5. He must not file a joint return with his spouse or be claiming a dependent of his own

6. He must not be the qualifying child of another taxpayer

He must have a US social security number or tax identification number (TIN)

In addition to the above requirements, to claim your boy/girlfriend's children, they must meet all of the above requirements and:
--- your boy/girlfriend must not be required to file a return,
--- he/she does not file a return claiming the children