A colleague recently pointed me to an article (Takeoff in Dense Fog…) at boldmethod that argues that the IFR takeoff minimums listed on IFR charts apply, even to operations under Part 91, if you accept a clearance that includes a DP, ODP, or SID.
But in most cases, these IFR takeoff minimums do not apply to pilots flying under Part 91, EXCEPT if the pilot is assigned and accepts a published departure procedure that includes takeoff minimums.
boldmethod 12/30/2017
The article references 14 CFR § 91.175 Takeoff and landing under IFR, specifically the standard takeoff minimums listed in paragraph (f).
Paragraph (f)(1) of the regulation notes:
Unless otherwise authorized by the FAA, no pilot may takeoff from a civil airport under IFR unless the weather conditions at time of takeoff are at or above the weather minimums for IFR takeoff prescribed for that airport under part 97 of this chapter.
But the text that introduces paragraph (f) also specifically notes that:
This paragraph applies to persons operating an aircraft under part 121, 125, 129, or 135 of this chapter.
14 CFR § 91.175 Takeoff and landing under IFR
In other words, those minimums do not apply to operations under Part 91. As explained in the Landis Letter (1999) [PDF], a letter of legal interpretation from an attorney at the FAA:
…Section 91.175(f) further states that no pilot operating an aircraft under Parts 121, 125, 127, 129, or 135…may takeoff from a civil airport under IFR unless the weather conditions are at or above the weather minimum for IFR takeoff prescribed for the airport under Part 97.
My research has indicated that when section 91.116(c)–the predecessor to the current section 91.175–was first proposed for adoption in 1967, this issue arose and was addressed as follows: Several comments indicated that section 91.116(c) of the proposal could be interpreted to apply civil airport takeoff minimums to aircraft operators other than those operating under Part 121, 129, or 135. As this result was not intended [underline added], the language of the paragraph as adopted herein has been changed to make it clear that the minimums apply to aircraft operating under 121, 129, or 135 (32 FR 13909 Oct 6, 1967). Accordingly, I am constrained to agree with your analysis that a Part 91 flight can depart in weather conditions less than the minima prescribed for the SID.
Note that the letter above does not mention Part 91K, which applies to Fractional Ownership Operations. Those operations are subject to additional constraints described in the management specifications FAA approves for each operator.
And that legal interpretation does not address the wisdom of departing in LIFR–perhaps zero-zero conditions–even if a close reading of the regulations allows it when you’re operating under Part 91.
Many experts recommend that you depart IFR only if weather at the airport you’re leaving is at least at or above the minimums for an approach at that airport, should you need to return quickly after takeoff. Some folks go further and use the circle-to-land minimums as their baseline. Or they plan for a takeoff alternate, which also isn’t required for Part 91 operations, but is always prudent.
BruceAir, LLC 
A general interpretation of Part 61 and re-affirmed by the Lander Letter, zero-zero take-offs are allowed for Part 91 operations. However it is not 100% true. Part 91 subchapter K requires a Management Specifications document approved by the FAA stating what is and what is not allowed for any particular fractional company. Mgt Specs may well include take-off restrictions. (Management Specifications is analogous to Part 135 / 121 Operational Specifications.)
Regardless of legality, wisdom and safety of a zero-zero take-offs, if there is an accident, the FAA can always resort to the catch all regulation: 91.13 “Careless or Reckless Operation”.
FAA can always place limitations on operations via operational specifications, limitations in an AFM, and via other means. And 91.13 is indeed a path to a violation when other, more specific, regulations don’t apply directly. The boldmethod article, however, is misleading and incorrectly asserts that accepting a clearance changes the IFR takeoff requirements for Part 91 operators. But I added text to note the Part 91K requirement.