Short answer. As a rule, yes. A substitute takes subject to the same charges and conditions imposed on the instituted heir. There are two exceptions: where the testator expressly provided otherwise, and where the charge or condition was personal to the original heir and could apply only to him.
What the law says
The substitute shall be subject to the same charges and conditions imposed upon the instituted heir
Civil Code, Article 862 — The Substitute Takes the Same Burdens. Read the full provision →
The substitute steps into the same shoes
Article 862 lays down the default: the substitute shall be subject to the same charges and conditions imposed upon the instituted heir. A substitution is not a fresh, unencumbered gift. The substitute receives the share as the testator shaped it, burdens and all, so if the original heir was to take on condition of doing something, or subject to a duty to pay a legacy or maintain a person, those attach to the substitute in the same way. The testator's overall plan for that share carries through the substitution rather than being stripped down when the backup takes.
Why the burdens follow the share
The logic is that the charge or condition was the testator's design for the property, not merely a personal arrangement with the first heir. When you place a duty on a share of your estate, you generally mean it to bind whoever ends up taking that share, because the point was to benefit some third person or to attach a requirement to the gift. Letting the burden fall away simply because a substitute inherited instead of the primary heir would frustrate that purpose, so the law keeps the conditions fastened to the share as it moves.
Where the burdens do not carry
The article names two exceptions. The first is an express provision by the testator to the contrary: you may free the substitute from a charge if you say so in the will. The second is where the charge or condition is personally applicable only to the instituted heir, tied to something only that particular person could do or be. A condition that the first heir personally care for a relative, or that turns on a circumstance unique to him, cannot sensibly bind a different person, so it does not pass to the substitute along with the share.
Read the will for both the burden and the exceptions
So whether a substitute inherits the conditions turns on the will's own terms. Look first at what charges and conditions were placed on the original heir, then at whether the testator expressly released the substitute from any of them, and finally at whether any condition was so personal to the first heir that it can attach to no one else. If none of the exceptions applies, the substitute takes the share exactly as burdened. A lawyer reading the institution and the substitution together can tell you which conditions the substitute actually bears.