Can an LP invest out of an LLC that an IRA owns?

From a tax perspective any K-1 income received from the fund would be tax deferred (assuming regular IRA) until the money is withdrawn from the IRA at which time it would be taxed at ordinary income rates. In effect the original character of the income would be lost so no long term capital gains rates or QSBS treatment if applicable. This is on top of the restricted transactions mentioned by Hans. 

Self-directed IRAs can be used to purchase LP interests.  These self-directed IRAs can be controlled by a custodian or managed by the IRA holder through a special purpose limited liability company which the IRA owns. LPs who want to invest through their IRAs should consult with their custodian or their tax professional who set up the self-directed IRA in order to make sure that they comply with complex regulations.  It is important to note that the IRS has restricted certain transactions between self-directed IRAs and persons related to the IRA holder that may be considered self-dealing or involve conflict of interest transactions.  If the IRA holder is related to a person who manages the fund, this should be carefully considered so as to comply with the regulations.