What instruments are/have been designed for tuning for just intonation (throughout history)? I'm looking for things that that are inherent in construction such as tone holes and fret placements.
The simple, short answer is that practically all families of instruments started out with just intonation and continued that way well into the 19th century (the 1800s).
The exception is keyboard instruments, and fretted instruments tuned in fourths, such as the guitar, and before it the lute and the viola da gamba. Keyboard and guitar-family instruments started in meantone tuning in the 1400s and 1500s. They evolved to employ various temperaments based on "compromises" of meantone tuning which eventually evolved into the use of 12-tone equal temperament centuries later.
In the late 19th century, various technologies were developed that enabled brass, reed and woodwind instruments to expand well beyond the just-intoned notes of the overtone series. For instance, trumpets and French horns were augmented with valves connecting up extra tubing tuned to an alternate and different series of overtones useable to produce additional notes.
The modern un-fretted bowed string instruments; the violin, viola, cello and bass, can and do play in just intonation and can modulate to any key center and maintain just intonation. Modern brass instruments without valves, such as the bugle, play in just intonation in one key only. The trombone, in use, is like a bugle that can adjust the length of its tubing to produce just intonation in different key centers.
If others here would like to elaborate on further details or exceptions or unusual circumstances with various instrument families, please post further comments.
And as @Pat Muchmore has pointed out, a choir of human voices, today, can and do sing in just intonation, provided that they are not being accompanied by a piano, other keyboard instrument, or guitar. If a choir sings accompanied by a modern piano or organ, the singers unconsciously tune the pitches they are singing to the 12-tone equal temperament of the piano.
If you want to know about fret placement on guitar-family instruments, please see the answers I gave to this question in which I provided information about modern guitars which can play in just intonation or various types of meantone temperament.