What things do I only get a limited opportunity to take photos of?

Some spoilers, from what I can recall:

• The master sword before you pull it out (you can always take a selfie later but it won't be in the rock anymore)
• All the base game bosses (blights, divine beast, Master Kohga, and ganon)
• The bow of light used at the end boss (You have to take a selfie with this one I think)
• All the DLC special enemies and the one hit obliterator (I think you can repeat the challenge to get more pictures?)
• According to the wiki: in Masters Mode you can possibly miss the Lynel Crusher, Mighty Lynel Crusher, and Lynel Spear as the enemies scale out of using those weapons.
• Ancient arrows (there's only a few places where you can find one laying around or in a breakable item) (see comment)

Note, if you do take a picture of the item you will never be able to buy the compendium version that's on sale, which may be of a better quality (like the bow of light).

Building on ||'s answer, information on the Breath of the WIld Object Map, and my own experience in filling the Compendium with pictures taken of objects in the game, below are the permanently-missable Compendium Photos in Breath of the Wild.

Creatures:

• There are no creatures that disappear permanently from the game.

Monsters:

• Guardian Sentries -- These only appear during the quest to board Vah Rudania. If you don't snap them during your climb, you'll never see them again.

• Cursed Bokoblins -- I might be wrong on this, but I've never encountered them outside of a Divine Beast. When I've gone looking after having completed all the Divine Beasts, all the Malice Mouths I could find only spewed Cursed Moblins and Cursed Lizalfos.

• The Blight Ganons (Thunderblight, Waterblight, Windblight, and Fireblight) are inaccessible after their respective Divine Beasts are completed, unless you have the Champion's Ballad DLC.

• Master Khoga cannot be refought, even with the DLC.

Materials:

• There are no materials that disappear permanently from the game.

Equipment:

• The Lynel Crusher, Lynel Spear (Normal Mode Only), Mighty Lynel Crusher, and Mighty Lynel Spear eventually are removed from the game as nearly all of the Lynels go Silver or higher.

• The Kite Shield is only found in chests, and as a Minigame Award. Open all the chests and win the one on the cabin wall in Selmie's Shield Surfing, and they'll all be gone.

Treasures:

• There are no treasures that disappear permanently from the game.