As stated by Dietrich Burde, it is known that there is no solution for $n\equiv \pm 4 \pmod{9}$, and conjectured that there are infinitely many solutions otherwise.

A cryptic aspect is that it is not even known that there exists one solution for all $n \not\equiv \pm 4 \pmod{9}$.

Today the smallest number for which the problem is open is $n=114$.

Here is a (non-exhaustive) history of the latest solutions found for $n \le 100$ (see here and there):

(1960s)

- $87 = 4271^3 – 4126^3 – 1972^3$
- $96 = -15250^3 + 13139^3 + 10853^3$
- $91 = 83538^3 – 67134^3 – 65453^3$
- $80 = -112969^³ + 103532^³ + 69241^³$

(1990s)

- $39 = -159380^³ + 134476^³ + 117367^³$
- $75 = – 435203231^³ + 435203083^³ + 4381159^³$
- $84 = 41639611^³ – 41531726^³ – 8241191^³$

(2000s)

- $30 = 2220422932^3 – 2218888517^3 – 283059965^3$
- $52 = -61922712865^3 + 23961292454^³ + 60702901317^3$
- $74 = −284650292555885^3 + 66229832190556^3 + 283450105697727^3$

(2019)

- $33 = 8866128975287528^3 - 8778405442862239^3 -2736111468807040^3$
- $42 = 80435758145817515^3 - 80538738812075974^3 + 12602123297335631^3$
- $3 = 569936821221962380720^3 - 569936821113563493509^3 - 472715493453327032^3$

*Remark*: for $n \le 1000$, the problem is still open only for $114$, $165$, $390$, $579$, $627$, $633$, $732$, $906$, $921$, and $975$ (see this paper and this paper, and also this).

Numberphile's videos:

all$x, y, z$ are positive, though I think this may serve as a little bit of help: $$\begin{align} 1 &= (9t^3 + 1)^3 + (9t^4)^3 + (-9t^4 - 3t)^3 \\ 2 &= (6t^3 + 1)^3 + (-6t^3 - 1)^3 + (-6t^2)^3 \end{align}$$ or forbigsolutions for $n = 1$: $$1 = (1 - 9t^3 + 648t^6 + 3888t^9)^3 + (-135t^4 + 3888t^{10})^3 + (3t - 81t^4 - 1296t^7 - 3888t^{10})^3$$ $\endgroup$ – Feeds Oct 27 '17 at 5:38