Can every true theorem that has a proof be proven by contradiction?
In classical logic, the answer is yes. Take any theorem $T$ and any proof $P$ for $T$. Now write the following proof:
If $\neg T$:
[Write $P$ here.]
Thus $T$.
Thus a contradiction.
Therefore $\neg \neg T$, by negation introduction.
Thus $T$, by double negation elimination.
One may object that this proof is essentially the same as $P$, and is just wrapped up. That is true, but it is a perfectly legitimate proof of $T$, even if it is longer than $P$, and it is indeed of the form of a proof by contradiction. A natural question that arises is whether the shortest proof of $T$ is a proof by contradiction. That is a much harder question to answer in general, but there are some easy examples, at least for any reasonable natural deduction system.
For instance, the shortest proof of "$A \to A$" for any given statement $A$ is definitely not a proof by contradiction but rather just:
If $A$:
$A$.
Therefore $A \to A$, by implication introduction.
On the other hand, the shortest proof of "$\neg ( A \land \neg A )$" for any given statement $A$ is definitely a proof by contradiction:
If $A \land \neg A$:
$A$, by conjunction elimination.
$\neg A$, by conjunction elimination.
Thus a contradiction.
Therefore $\neg( A \land \neg A )$.
The first part of this post shows that the shortest proof by contradiction is at most a few lines longer than the shortest proof, but nothing much else interesting can be said about the shortest proof unless...
Well what if we do not allow the use of double negation elimination? If you have only the other usual rules (the first-order logic rules here but excluding ¬¬elim and including ex falso), then the resulting logic is intuitionistic logic, which is strictly weaker than classical logic, and cannot even prove the law of excluded middle, namely "$A \lor \neg A$" for any statement $A$. So if you instead ask the more interesting question of whether every true theorem can be proven in intuitionistic logic, then the answer is no.
Note that intuitionistic logic plus the rule "$\neg A \to \bot \vdash A$" gives back classical logic, and one could say that this rule embodies the 'true principle' of proof by contradiction, in which case one can say that some true theorems require the use of a proof by contradiction somewhere.
If you can prove a statement $A$ directly, you can prove it by contradiction. Just assume $\lnot A$, perform your proof of $A$, note the contradiction, and derive $A$.