Do I need to replace the foundation under porch posts when replacing them?
I would say this home is C 1920. But I guess it could be built more recently like and older home.  It's possible those columns are not so load bearing.  The joists above the porch could be cantilevered. Like.. But even if load bearing I don't think it is that big of a deal. Just support them well under the porch. http://stairs4u.com/glossary/images/cantilevered-floor-joist-fr.jpg

I did pull out a column like this on a couple houses. I did use a jack to lift the corner of the porch ceiling a little to be able to pull the post/columns out. Once you jack it up you just jam a 2x4 under and going down to a support board on the ground.

If the corner of the porch has not sunk down much over the years, it seems what was there is working ok.  Or if you want a more substantial support you could pour a small square or round (they make tubes you can dig in then fill) foundation and embed a post base in to the concrete when wet. Like: https://resourcefulsupply.com/storage/products/107379-400x400-resizeupscale(1).jpg

You could put a concrete pier with wood top in a pad of poured concrete too. Like https://images.homedepot-static.com/productImages/87e13217-d9b6-4993-8dc9-9d855ba3000f/svn/cinder-blocks-100014754-64_1000.jpg

Sometimes those columns on the porch are not solid and just nailed to the deck of the porch. If they are solid 4x4 or 6x6 they could be nailed to the porch deck too.  They seem to have some finishing decorative wood around them.  You might be able to just use the same columns after removing them  then cut of a few inches of rotted material and reinstall on top of a 2x6x6 (or?) base that you could route the edges to be decorative. 

If unsure or uncomfortable or in earthquake area have an engineer look at it.  Probably best to get a permit too and the city inspector might be able to give you and idea or two to be at current standards.  Best of luck!