Seems like for a Raspberry Pi A+ (armv6, 32-bit) I have to use the non-armhf package, or armel package AND I have to apply a symbolic link fix. Your documentation should get an update for this as I was testing with a pure image of jessie to get this working.
FROM resin/rpi-raspbian:jessie
# Sources:
# - https://help.getsync.com/hc/en-us/articles/206178924-Installing-Sync-package-on-Linux
# - https://wiki.archlinux.org/index.php/Resilio_Sync#Installation
# Install tools
RUN apt-get update && apt-get install -y \
curl \
wget \
vim
# Install Resilio Sync
WORKDIR /opt/resilio/bin
RUN wget https://download-cdn.resilio.com/stable/linux-arm/resilio-sync_arm.tar.gz
RUN tar xf resilio-sync_arm.tar.gz
RUN ln -sf /lib/ld-linux-armhf.so.3 /lib/ld-linux.so.3 # Fix
EXPOSE 8888
# Need's more work to get the container to run with this
CMD ["./rslsync", "--webui.listen", "0.0.0.0:8888"]