"er war" is the past form (Präteritum) of the verb "sein" ("er ist") whereas "er sei" is Konjunktiv 1 of "sein".
Konjunktiv is used in the indirect speech and in conditionals.
In the example you give, you have to use Konjunktiv 1.
Konjunktiv 1:
ich sei, du seist, er sei, wir seien, ihr seit, sie seien
Konjunktiv 2 :
ich wäre, du wärst, er wäre, wir wären, ihr wärt, sie wären
Konjunktiv 1 is used in indirect speech but if it's form is equal to an other form of the verb use Konjunktiv 2 instead (so don't use "ihr seit" as konjunktiv form but "ihr wärt"). This Konjunktiv 2 has then exactly the same meaning as Konjunktiv 1. In other context it can bring some unreality or doubt into the sentence.
In spoken language Konjunktiv 1 is often totally replaced by Konjunktiv 2 or even Indikativ. But the the sentence could then mean something different.
if you use "war" in your example and continue maybe with "aber":
Die Ärztin hatte meiner Mutter gesagt, dass meine Leber kurz vor der Zirrhose war aber jetzt wieder gut sei.
you totally changes the meaning of the sentence. With "sei" it said that at the time the Doctor said it, he mend it. With "war ..." it says the the doctor said that it had been like this but is now, at the time he says it, different.
(Not that it would be better to use "gewesen sei" to achieve this meaning. Since ending the sentence directly after "war" could be interpreted as if you used "sei").
Compare:
Ich sage, dass ich hier bin.
I say that I'm here.
Ich sage, dass ich hier war.
I said that I was here.
Ich sagte, dass ich hier sei.
I said that I was here
Ich sagte, dass ich hier wäre.
I said that I was here.
This sentence could be interpreted as if you continued with "but I wasn't", but is often (in spoken language) used instead of the sentence with "sei".
Ich sagte, dass ich hier war.
I said that I was here or I said that I had been here, depending on the context.
Ich sagte, dass ich hier gewesen sei/wäre.
I said that I had been here (before).
with "wäre", it by be interpreted like the sentence with "wäre" above (but in past perfect of course).
(Note that there are more possibilities to say these things. For simplicity I always used "dass", which is bad style if you only use it.)
war
would meanhas been, is not
furthermore.
sei` undwäre
would state that it probably will come to a cirrhosis.