What does "pillow-plumping romance with the press" mean?
Solution 1:
It's nowhere near a "standard idiom", so strictly thinking I think this question might be off-topic (it's effectively literary criticism/interpretation).
Obama's romance is just an alternative to honeymoon [with the press] - a common metaphor attaching to public figures newly risen to the attention of the media.
Literally, pillow-plumping is just beating and fluffing up a pillow to make it more comfortable. When conjoined with the metaphoric press romance, it emphasises how the press have made his early years as president comfortable/easy. Ensuring her man has a nice soft place (breast, pillow) to lay his head is something an adoring lover might do in the first flush of new romance.
This "soft pillow" metaphor harks back to Obama's "monastic skills" as mentioned earlier in the passage. By cliched convention (and to some extent, historical fact), monks live ascetic/abstinent existences, in which context beds of nails, and hard stone pillows, are common images.
TL;DR: pillow-plumping here means indulgent and comforting.
Solution 2:
My understanding is that this is all about how, when you're reading a really good book in bed, you might thump your pillow to make it fatter, so that you can sit up in bed more comfortably. Thumping your pillow this way is called "plumping", because it makes your pillow plumper. So a "pillow-plumping" romance would be a romance novel that you would read in bed; more particularly, it's a gripping enough book that you would "plump up" your pillow and read it, in preference to switching off the light and going to sleep.
In the case of Obama, it's kind of a metaphor; the American people were eager to read about Obama, and everything he did that was reported favourably by the press. Now, they're not so eager any more - the pillow plumping has finished.
As worded in the Times article, pillow-plumping romance probably refers to the act of pillow plumping – how one lover might plump the pillow for the other, suggesting the great care, deference, attention, and infatuation that often accompany the early stages of a new romantic relationship. In other words, suggesting that the pillow-plumping romance is over is another way of saying that the (metaphorical) honeymoon is over.
Solution 3:
I’ve never heard pillow-plumping before, but it sounds like something they might do for you at a cosy little bed-and-breakfast, fluffing your pillow for you in the evening.