It looked out of place
My guess is that it simply looked out of place, either because it didn't match the rest of the decor (it was patterned on Slughorn's pyjamas, hardly the most common material used for an armchair), because he'd neglected to cover himself in the same dragonsblood spatter or was just too clean in general.
Cushions lay deflated, feathers oozing from slashes in their sides;
fragments of glass and china lay like powder over everything.
Dumbledore raised his wand even higher, so that its light was thrown
upon the walls, where something darkly red and glutinous was spattered
over the wallpaper. Harry’s small intake of breath made Dumbledore
Dumbledore is already looking for something bulky and incongruous
Dumbledore states that he knew that Slughorn was still in the house because of the lack of a dark mark and he would also be aware of his skill in transformation. Seeing that the armchair is the only thing mentioned that would be large enough to accommodate him, in hindsight it seemed the obvious choice.
‘What gave it away?’ he grunted as he staggered to his feet, still
rubbing his lower belly. He seemed remarkably unabashed for a man who
had just been discovered pretending to be an armchair. ‘My dear
Horace,’ said Dumbledore, looking amused, ‘if the Death Eaters really
had come to call, the Dark Mark would have been set over the house.’
The wizard clapped a pudgy hand to his vast forehead. ‘The Dark Mark,’
he muttered. ‘Knew there was something … ah well. Wouldn’t have had
time, anyway. I’d only just put the finishing touches to my upholstery
when you entered the room.’
Dumbledore can detect magic
Dumbledore later shows an ability to sense powerful magic. He detects a cleverly concealed entrance simply by murmuring a few words and touching it.
Harry did not ask how Dumbledore knew. He had never seen a wizard work
things out like this, simply by looking and touching; but Harry had
long since learned that bangs and smoke were more often the marks of
ineptitude than expertise.
And although I appreciate you're asking questions about the book and not the film, it's certainly worth pointing out it was a little easier in the cinematic version given that armchairs don't usually have feet sticking out from under them :-)