On Ubuntu 16.04, the command 

sage -t --all --optional=sage,optional,external

tests the following optional and external doctests:

Using 
--optional=bliss,cbc,ccache,cmake,dot2tex,external,gmpy2,lrslib,memlimit,mpir,normaliz,notedown,pandoc_attributes,pycosat,pynormaliz,python2,rst2ipynb,sage
External software detected for doctesting: 
ffmpeg,graphviz,gurobi,imagemagick,internet,latex,pandoc

and gives All tests passed except the following ones:

----------------------------------------------------------------------
sage -t --long src/sage/databases/findstat.py  # 1 doctest failed
sage -t --long src/sage/combinat/tutorial.py  # 1 doctest failed
sage -t --long src/sage/symbolic/integration/integral.py  # 1 doctest failed
sage -t --long src/sage/finance/stock.py  # 20 doctests failed
sage -t --long src/sage/combinat/designs/ext_rep.py  # 1 doctest failed
sage -t --long src/sage/symbolic/integration/external.py  # 3 doctests 
failed
sage -t --long src/sage/repl/load.py  # 1 doctest failed
sage -t --long src/sage/misc/persist.pyx  # 2 doctests failed
----------------------------------------------------------------------

Follow up at https://trac.sagemath.org/ticket/25536. New failures are 
copied below.

sage -t --long --optional=sage,optional,external 
src/sage/databases/findstat.py
**********************************************************************
File "src/sage/databases/findstat.py", line 1232, in 
sage.databases.findstat.FindStatStatistic.function
Failed example:
    findstat("Permutations", lambda pi: pi.length()).function()   # 
optional -- internet
Expected:
    ...
    <function <lambda> at ...>
Got:
    <function <lambda> at 0x7f72c8ce2c80>
**********************************************************************
1 item had failures:
   1 of   2 in sage.databases.findstat.FindStatStatistic.function
    [247 tests, 1 failure, 213.55 s]


sage -t --long --optional=sage,optional,external 
src/sage/combinat/tutorial.py
**********************************************************************
File "src/sage/combinat/tutorial.py", line 224, in sage.combinat.tutorial
Failed example:
    oeis([1,1,2,5,14])                            # optional -- internet
Expected:
    0: A000108: Catalan numbers: C(n) = binomial(2n,n)/(n+1) = 
(2n)!/(n!(n+1)!). Also called Segner numbers.
    1: A120588: G.f. satisfies: 3*A(x) = 2 + x + A(x)^2, with a(0) = 1.
    2: ...
Got:
    0: A000108: Catalan numbers: C(n) = binomial(2n,n)/(n+1) = 
(2n)!/(n!(n+1)!). Also called Segner numbers.
    1: A124302: Number of set partitions with at most 3 blocks; number of 
Dyck paths of height at most 4; dimension of space of symmetric polynomials 
in 3 noncommuting variables.
    2: A120588: G.f. satisfies: 3*A(x) = 2 + x + A(x)^2, with a(0) = 1.
**********************************************************************
1 item had failures:
   1 of 249 in sage.combinat.tutorial
    [248 tests, 1 failure, 13.69 s]

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