You're not going to get a consistent answer on this.
Some highly respected, purist makers will absolutely refuse to use sap anywhere in their furniture, saying it's not as strong, worm prone, or just not attractive.
On the other hand Black Walnut for example is almost always steamed to minimise the colour difference between sap and heart and then sold on a "sap no fault" basis. Then there are timbers like Yew, where there's so much ingrown bark that you'd struggle to make anything without including at least a bit of sap. Or there's tropicals like most Rosewoods where the sap is actually harder than most European timbers, and is now often incorporated into contemporary furniture designs to take advantage of the vivid colour distinction between the creamy sap and the purply black heart.
Personally I think you should take a pragmatic position, if the sap is firm and worm free for use in a centrally heated home (where worm isn't an issue) then it's your legitimate decision to use it or not.
Incidentally, I'm with Beau regarding your Ash, I think you're looking at Olive Ash/Ash heart.
Here are some rippled Ash boards I used recently in a furniture build, it's all heartwood, but the darker sections are streaks of Olive Ash (it's a result of a fungal attack in the growing tree). Some times the whole tree is stained the darker colour, other times it's a more localised. Again it's a personal call, but used carefully it can deliver a unique decorative effect to your furniture.
If you're making individually designed pieces of furniture then boards like this are another good reason why you should always inspect and buy your boards in person.