Short answer. His estate steps in. Under Article 1861, on the death of a limited partner his executor or administrator has all the rights of a limited partner for the purpose of settling the estate, and the power the deceased had to constitute his assignee a substituted limited partner. The estate is liable for all his liabilities as a limited partner.
What the law says
The estate of a deceased limited partner shall be liable for all his liabilities as a limited partner.
Civil Code, Article 1861 — Death of a Limited Partner. Read the full provision →
The estate inherits the rights
A limited partner's interest does not simply vanish when he dies — it passes into his estate, and Article 1861 gives his personal representative the standing to deal with it. On the death of a limited partner his executor or administrator shall have all the rights of a limited partner for the purpose of setting his estate. So the executor or administrator can exercise the deceased's rights as a limited partner — inspecting the books, demanding information and an account, collecting the share of profits, seeking the return of the contribution — to the extent needed to gather in and settle the estate.
And the power to substitute an assignee
The representative also inherits a specific power the deceased had: such power as the deceased had to constitute his assignee a substituted limited partner. A substituted limited partner is someone who takes an assignee's place as an actual limited partner, with the rights that go with membership, rather than merely receiving the economic benefit of the interest. If the deceased limited partner had the power — under the certificate — to make his assignee a substituted limited partner, his executor or administrator can exercise that power on the estate's behalf.
The estate keeps the liabilities too
The inheritance is not all upside. The article ends: the estate of a deceased limited partner shall be liable for all his liabilities as a limited partner. So whatever the deceased owed the partnership in that capacity — an unpaid contribution he had promised, a shortfall between what he actually put in and what the certificate stated, property he held in trust for the firm — passes to the estate as a liability. The heirs do not take the interest free of those obligations.
For the estate of a limited partner
If you are administering the estate of someone who was a limited partner, you step into his shoes for that interest: you can seek information and an account from the partnership, claim the profits and the return of the contribution, and, where the certificate allowed it, substitute an assignee as a limited partner. Handle the interest as an asset of the estate. But account for the other side too — the estate answers for the deceased's limited-partner liabilities, so identify any unpaid or over-stated contribution before distributing. Get the partnership's records early, because both the value of the interest and the extent of the liabilities come from them.