Which table/dataset do I need to know which city/place has the more Asian male & white female couples (greater number/a higher proportion)?

To my knowledge, Glenn is correct that no published census or ACS tables provide counts of couples by their individual races/ethnicities.

You could generate counts like this using ACS microdata. The public use microdata gives the survey responses for a sample of households, including the race & ethnicity of every household member. You could use this info to identify which households have a householder who is Asian male or white female with a spouse or cohabiting partner who is (respectively) white female or Asian male. That would limit your analysis to couples who head a shared household (excluding group quarters population and non-householder couples), but it would include most couples.

The bigger challenge for your question is that the public use microdata don't identify cities/places. The smallest identified areas are Public Use Microdata Areas (PUMAs), which must have at least 100,000 residents each, so you wouldn't be able to identify smaller cities.

IPUMS USA provides ACS microdata samples with tutorials to get you started and a tool to summarize microdata online. It also identifies cities and metro areas, when possible, based on the available PUMA information. The Census Bureau provides similar tooling through "MDAT", but that tool doesn't identify cities or metro areas.

Good point, I should have mentioned microdata. Is it available for areas smaller than PUMAs in the census research centers? That might not be practical for the OP.

Yes, that's true. If you apply for and receive access to through an FSRDC, you could use the complete ACS microdata, which identifies each household's block, potentially even the street address if you get permission to use that, too. It still might not be a large enough sample to produce good estimates for smaller cities/places. You'd at least need to pool multiple years' samples, and for the smallest places, that may still not be enough.

Yes, that's true. If you apply for and receive access to through an FSRDC, you could use the complete ACS microdata, which identifies each household's block, potentially even the street address if you get permission to use that, too. It still might not be a large enough sample to produce good estimates for smaller cities/places. You'd at least need to pool multiple years' samples, and for the smallest places, that may still not be enough.

This a little advertisement for the Small Area Estimation package that is in my earlier post. 

The variable Asian male & white female couples can be constructed using PUMS data so the SAE package will produce tract level data with this variable.

You can easily get places geo by aggregating tract output synthetic microdata from the package.

Dave