Let's took at the different possible assumptions.
If you assume a unique classical universe with one-way causality, then obviously, the fact you only throw heads is just luck, it doesn't say anything about the initial throw. You could promote hypothesis like "the coin is biased, because the ones running that experiment are sadistic who like to see my fear every time the coin is tossed", and maybe from that you could change your estimate of the first coin being biased, depending of your understanding of human psychology.
If you assume MWI it gets more complicated, and depends if the coin tossing is "quantum-random" or not. A usual coin tossing is not "quantum-random", the reasons behind the coin ending heads or tails are in classical physics, well above the level of "quantum noise". So in all the existing worlds (or a very, very large majority of them), the toss will give the same results, so you're back to the first case.
If you assume MWI and the coin toss are "quantum random", then after two tosses (not counting the initial), you've 8 outcomes : HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. You can only experience H and THH. On the 8 copies of you, 3 will be dead, 5 alive. But for there are two copies seeing HH, you can't tell apart THH and HHH, both have p=1/2. p(HH | I'm alive) = 2/5, but p(T | *HH and I'm alive) is still 1/2.
Now, if you assume the first coin was not quantum, but the subsequent are ... there is no longer 8 worlds having the same Born probabilities (the same "level of existence") but only two sets of 4 worlds, each set having a 1/2 probability of existing. In one set of world, there will be just one copy of you seeing "HH", on the other, there will be 4 copies of you seeing all the possible outcomes.
Then we can rephrase the problem in a way that doesn't involve the anthropic principle, making it a more classical problems of probabilities. We take 5 similar pieces of paper. Two are written HH, the others HT, TH and TT. One HH is put aside, then each are folded. Then someone tosses a coin. If it's tails, you are given the lone HH paper. If it's heads, you're giving one at random among the 4 papers. You open your paper, and it's HH. What's the probability the coin landed tails ? Well, it's 4/5, not 1/2.
So, my answer is : if MWI holds, and the first coin is not quantum random, but the later are quantum random, you should consider the 1000 heads (and even stop much before) to be strong evidence towards "the initial was tails". If none or all of the coins are quantum random, or you don't believe in MWI, you shouldn't.
Closely related to: How Many LHC Failures Is Too Many?
Consider the following thought experiment. At the start, an "original" coin is tossed, but not shown. If it was "tails", a gun is loaded, otherwise it's not. After that, you are offered a big number of rounds of decision, where in each one you can either quit the game, or toss a coin of your own. If your coin falls "tails", the gun gets triggered, and depending on how the original coin fell (whether the gun was loaded), you either get shot or not (if the gun doesn't fire, i.e. if the original coin was "heads", you are free to go). If your coin is "heads", you are all right for the round. If you quit the game, you will get shot at the exit with probability 75% independently of what was happening during the game (and of the original coin). The question is, should you keep playing or quit if you observe, say, 1000 "heads" in a row?
Intuitively, it seems as if 1000 "heads" is "anthropic evidence" for the original coin being "tails", that the long sequence of "heads" can only be explained by the fact that "tails" would have killed you. If you know that the original coin was "tails", then to keep playing is to face the certainty of eventually tossing "tails" and getting shot, which is worse than quitting, with only 75% chance of death. Thus, it seems preferable to quit.
On the other hand, each "heads" you observe doesn't distinguish the hypothetical where the original coin was "heads" from one where it was "tails". The first round can be modeled by a 4-element finite probability space consisting of options {HH, HT, TH, TT}, where HH and HT correspond to the original coin being "heads" and HH and TH to the coin-for-the-round being "heads". Observing "heads" is the event {HH, TH} that has the same 50% posterior probabilities for "heads" and "tails" of the original coin. Thus, each round that ends in "heads" doesn't change the knowledge about the original coin, even if there were 1000 rounds of this type. And since you only get shot if the original coin was "tails", you only get to 50% probability of dying as the game continues, which is better than the 75% from quitting the game.
(See also the comments by simon2 and Benja Fallenstein on the LHC post, and this thought experiment by Benja Fallenstein.)
The result of this exercise could be generalized by saying that counterfactual possibility of dying doesn't in itself influence the conclusions that can be drawn from observations that happened within the hypotheticals where one didn't die. Only if the possibility of dying influences the probability of observations that did take place, would it be possible to detect that possibility. For example, if in the above exercise, a loaded gun would cause the coin to become biased in a known way, only then would it be possible to detect the state of the gun (1000 "heads" would imply either that the gun is likely loaded, or that it's likely not).