Can you count the number of rectangles?

JavaScript (ES6), 176 171 bytes

g=a=>Math.max(...b=a.map(a=>a.length))-Math.min(...b)?``:f(a);f=
a=>a.map((b,i)=>[...b].map((_,j)=>n+=a.join`
`.split(eval(`/\\+(?=[-+]{${j}}\\+[^]{${l=b.length+~j}}([|+].{${j}}[|+][^]{${l}}){${i}}\\+[-+]{${j}}\\+)/`)).length>>1),n=0)|n

<textarea rows=8 cols=8 oninput=o.textContent=g(this.value.split`\n`)></textarea><pre id=o>

Takes input as an array of strings of equal length. Explanation: Creates a series of regular expressions that match rectangles of all possible widths and heights (and some impossible widths and heights, but that's code golf for you) and counts how many matches they all produce. Because there's a capturing group in the regexp, split returns 2n+1 for n matches, so I right shift by 1 to get the number of matches, as that saves a byte over making the group non-capturing.

Grime, 31 28 bytes

T=\+[+\-]*\+/[+|]/+$
n`T&To2

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Takes input in ASCII format.

Explanation

Grime's syntax is very close to regular expressions. Each line defines a pattern that may or may not match a rectangle of characters. T matches a rectangle whose top row and left column look valid.

T=\+[+\-]*\+/[+|]/+$
T=                    Define T as
  \+[+\-]*\+          a row that matches this regex
            /         and below that
             [+|]/+   a column of + or |
                   $  with anything to its right.

The second row is the "main program".

n`T&To2
n`       Print number of rectangles that match
  T      the pattern T
   &     and
    To2  T rotated 180 degrees.

J, 103 95 86 80 76 70 bytes

[:+/@,]*/@('-|++'*/@(e.,&'+')~&>]({.,{:)&.>@;|:;{.;{:);._3"$~2+$#:i.@$

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Takes input as an array of strings with trailing spaces (so that each string is the same size). Uses the full subarray operator ;._3 to iterate over every possible subarray size larger than 2 x 2, and counts the subarrays that are valid rectangles. Completes all test cases almost instantly.