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Terminology:

• 
  Cell: the LSTM unit containing input, forget, output gates and the hidden hT and cell state cT.


• 
  Hidden units/memory: How far back in time the LSTM is "unrolled". A hidden unit is an instance of the cell at a particular time.


• A hidden unit is parameterized by [wT, cT, hT-1]: The gate weights for the current hidden unit, the current cell state, and the last hidden unit's output. Where wT represents input, output, forget gate weights.

An LSTM maintains separate gate weights wT for each hidden unit. This way it can treat different points in time of a sequence differently.

Let's say an LSTM has 3 hidden units so has gate weights w1, 2, w3 for each of them. Then a sequence x1, x2,...xN comes through. I am illustrating the cell as it transitions over time:

@ t=1

xN....x3    x2    x1



                 [w1, c1, h0]



                     (c2, h1)



@ t=2

xN....x4    x3    x2                   x1



                 [w2, c2, h1]



                     (c3, h2)



@ t=3

xN....x5    x4    x3                   x2    x1



                 [w3, c3, h2]



                     (c4, h3)

But what happens at t=4? The LSTM only has memory, therefore gate weights, for 3 steps:

@ t=4

xN....x6    x5    x4                   x3    x2    x1



                 [w?, c3, h2]



                     (c4, h3)

What weights are used for x4 and all the following inputs? In essence, how are sequences that are longer than an LSTM cell's memory treated? Do the gate weights reset back to w1, or do they remain static at their latest value wT?

Edit: My question is not a duplicate of the LSTM inference question. It is asking about multi-step prediction from inputs. However, I am asking about what weights are used over time for sequences that are longer than the internal hidden cell states. The question of weights is not addressed in that answer.

The gates are a function of the weights, the cell state and the hidden state.
The weights are fixed.

Consider the equation for the forget gate $f_t$:
$$f_t = sigma(W_f cdot [h_{t-1}, x_t]+b_f)$$
The forget gate uses the new data $x_t$ and the hidden state $h_{t-1}$, but $W_f$ and $b_f$ are fixed. This is why the LSTM only needs to keep the previous $h$ and the previous $c$.

More information:
http://colah.github.io/posts/2015-08-Understanding-LSTMs/

Expanding a little bit what Sycorax said, the basic recurrent cell is something like

$$ h_t = tanh(W_{hh}h_{t-1} + W_{xh}x_t) $$

It is a function of previous hidden state $h_{t-1}$ and current input $x_t$, and returns current hidden state $h_t$. Same applies to LSTM cell that is a special kind of RNN.

So it does not "look" directly at any input in the past, the information from the past is passed only through the hidden states $h_t$. What follows, in theory all the history to some extent contributes to $h_t$. If you have multiple such cells, then they do not look directly at different points in time, but just use different weights. Of course, it can be the case that some of the hidden states will carry more information from points that are further in time, as compared to other hidden states, but the kind of information that is carried is something that is learned from the data, rather then forced.